Alliances in graphs: Parameters, properties and applications—A survey

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Alliances in graphs: Parameters, properties and applications—A survey Kahina Ouazine, Hachem Slimani ∗, Abdelkamel Tari LIMED Laboratory, Computer Science Department, University of Bejaia, 06000 Bejaia, Algeria Received 19 October 2016; received in revised form 15 May 2017; accepted 24 May 2017 Available online xxxxx

Abstract In practice, an alliance can be a bond or connection between individuals, families, states, or entities, etc. Formally, a non-empty set S of vertices of a graph G is a defensive k-alliance (resp. an offensive k-alliance) if every vertex of S (resp. the boundary of S) has at least k more neighbors inside of S than it has outside of S. A powerful k-alliance is both defensive k-alliance and offensive (k+2)alliance. During the last decade there has been a remarkable development on these three kinds of alliances. Due to their variety of applications, the alliances in its broad sense have received a special attention from many scientists and researchers. There have been applications of alliances in several areas such as bioinformatics, distributed computing, web communities, social networks, data clustering, business, etc. Several k-alliance numbers have been defined and a huge number of theoretical (algorithmic and computational) results are obtained for various graph classes. In this paper, we present a survey which covers a number of practical applications of alliances and the vast mathematical properties of the three types of k-alliances by giving a special attention to the study of the associated k-alliance (partition) numbers for different graph classes. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Defensive (offensive powerful) k-alliance; Boundary k-alliance; Partitioning of graphs; k-alliance (partition) number; Graph class

Contents 1.

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Introduction .................................................................................................................................................................................................. 1.1. Historical view and applications of alliances......................................................................................................................................... 1.2. Terminology and definitions ............................................................................................................................................................... Defensive k-alliances in graphs ....................................................................................................................................................................... 2.1. Study of defensive k-alliance numbers for various graphs classes............................................................................................................ 2.1.1. General graphs .................................................................................................................................................................. 2.1.2. Tree graphs ....................................................................................................................................................................... 2.1.3. Planar graphs .................................................................................................................................................................... 2.1.4. Complete graphs ............................................................................................................................................................... 2.1.5. Bipartite graphs and complete bipartite graphs ......................................................................................................................

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Peer review under responsibility of Kalasalingam University. ∗ Corresponding author. Fax: +213 34 813709.

E-mail addresses: [email protected] (K. Ouazine), [email protected] (H. Slimani), [email protected] (A. Tari). http://dx.doi.org/10.1016/j.akcej.2017.05.002 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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2.1.6. Regular graphs .................................................................................................................................................................. 2.1.7. Cycle graphs ..................................................................................................................................................................... 2.1.8. Path graphs ....................................................................................................................................................................... 2.1.9. Line graphs....................................................................................................................................................................... 2.1.10. Cartesian product graphs .................................................................................................................................................... 2.2. Study of defensive k-alliance partition numbers for some graph classes ................................................................................................... 2.2.1. General graphs .................................................................................................................................................................. 2.2.2. Tree graphs ....................................................................................................................................................................... 2.2.3. Regular graphs .................................................................................................................................................................. 2.2.4. Cartesian product graphs .................................................................................................................................................... 2.2.5. Partitioning a graph into boundary defensive k-alliances ........................................................................................................ Offensive k-alliances in graphs ........................................................................................................................................................................ 3.1. Study of offensive k-alliance numbers for various graph classes.............................................................................................................. 3.1.1. General graphs .................................................................................................................................................................. 3.1.2. Tree graphs ....................................................................................................................................................................... 3.1.3. Planar graphs .................................................................................................................................................................... 3.1.4. Complete graphs ............................................................................................................................................................... 3.1.5. Bipartite graphs and complete bipartite graphs ...................................................................................................................... 3.1.6. Regular graphs .................................................................................................................................................................. 3.1.7. Cycle graphs ..................................................................................................................................................................... 3.1.8. Path graphs ....................................................................................................................................................................... 3.1.9. Line graphs....................................................................................................................................................................... 3.1.10. Cartesian product graphs .................................................................................................................................................... 3.2. Study of offensive k-alliance partition numbers..................................................................................................................................... 3.2.1. General graphs .................................................................................................................................................................. 3.2.2. Cartesian product graphs .................................................................................................................................................... 3.2.3. Circulant graphs — C R(n, 2).............................................................................................................................................. go 3.2.4. Relations between ψk (G) and k ......................................................................................................................................... 3.2.5. Partition number and chromatic number ............................................................................................................................... Powerful k-alliances in graphs......................................................................................................................................................................... 4.1. Study of powerful k-alliance numbers for various graph classes .............................................................................................................. 4.1.1. General graphs .................................................................................................................................................................. 4.1.2. Tree graphs ....................................................................................................................................................................... 4.1.3. Planar graphs .................................................................................................................................................................... 4.1.4. Complete graphs ............................................................................................................................................................... 4.1.5. Complete bipartite graphs ................................................................................................................................................... 4.1.6. Regular graphs .................................................................................................................................................................. 4.1.7. Cycle graphs ..................................................................................................................................................................... 4.1.8. Path graphs ....................................................................................................................................................................... 4.1.9. Cartesian product graphs .................................................................................................................................................... 4.2. Study of powerful k-alliance partition numbers ..................................................................................................................................... 4.2.1. General graphs .................................................................................................................................................................. 4.2.2. Cartesian product graphs .................................................................................................................................................... 4.2.3. Partitioning a graph into boundary powerful k-alliances ......................................................................................................... Conclusion and discussion .............................................................................................................................................................................. Acknowledgments ......................................................................................................................................................................................... Appendix ...................................................................................................................................................................................................... References ....................................................................................................................................................................................................

10 10 10 11 11 12 16 16 17 17 17 19 19 19 21 21 22 22 22 23 23 23 23 27 27 28 28 29 29 29 30 30 31 32 32 32 33 33 33 33 35 35 35 35 37 37 37 38

1. Introduction 1.1. Historical view and applications of alliances The word alliance can be defined as a union or association formed for mutual benefit, for example it can be: a formal agreement or treaty between two or more nations to cooperate for specific purposes, a merging of efforts or interests by persons, families, states, or organizations. The study of alliances in graphs is first investigated by Kristiansen et al. [1] by defining different types of alliances that have been extensively studied in the last decade. These types of alliances are called defensive alliances [2,3], offensive alliances [4,5] and dual or powerful alliances [6,7]. A generalization of these alliances called k-alliances (or r -alliances) introduced by Shafique and Dutton [8,9] have received a special attention in recent years. In this setting, there have been definitions of many and various parameters which have been Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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studied widely in the case of defensive (offensive, powerful) k-alliances and for different graphs classes. The study of the alliances (in its broad sense) is motivated by interesting applications in several areas. The research work of Kristiansen et al. [1] is motivated by the alliance of nations in war for mutual support. They considered two cases: the first which corresponds to a defensive alliance is realized when nations are obligated to join forces if one or more of them are attacked and the second is an offensive alliance which is used as a mean of keeping the peace; as an example of illustration, the action of NATO troops in a war-torn country is given. Thus in the corresponding graph of this situation, the vertices represent the nations and the edges correspond to possible relations (of either friendship or hostility) between them. Essentially, Kristiansen et al. [1] studied the mathematical properties of defensive alliances in graphs. Haynes et al. [10] studied structural characteristics of a class of biomolecules in the bioinformatics arena by involving several graphical invariants based on domination numbers. In fact, they studied the applicability of graphs in the analysis of secondary RNA structure. They used graph-theoretic trees as a modeling method to represent secondary RNA motifs. Specifically, they utilized five domination parameters including the global defensive (−1)alliance number, that are highly sensitive to the structural changes of small ordered trees, to identify which trees of orders seven and eight are RNA-like in structure. With this study, it is shown that graphical invariants, which aid in the optimization of computer and electrical networks, are useful and serve as an interesting tool for genomic and proteomic predictions. In distributed computing, one of the central problems is how to deal with failures. Flocchini et al. [11], Peleg [12], Srimani and Xu [13], and Xu and Srimani [14] studied the fault tolerance of distributed computing and communication networks. Generally, a distributed system is represented by a graph where the vertices represent the processors and the edges correspond to different communications between them. Thus by using the process of local majority voting in graphs, processors are partitioned into two alliances. Furthermore, Srimani and Xu [13], and Xu and Srimani [14] designed self-stabilizing fault tolerant distributed algorithms for the global defensive (offensive) alliances in a given arbitrary graph. Flake et al. [15] defined a community on the web as a set of web pages that link (in either direction) to more web pages in the community than to pages outside of the community. With this definition, the defensive alliances represent exactly the mathematical model of such web communities. Szab¨o and Cz´ar´an [16] and Kim and Liu [17] have studied defensive alliances in cyclical interaction models of six mutating species which represent generalizations of the Rock–Scissors–Paper game. The alliances are also used in business and social networks in order to achieve common objectives by partners. In this context, Dickson and Weaver [18] studied the interaction between the firm size and the level of national R&D intensity to determine if it would be interesting for an SME (small and medium enterprise) to form a strategic alliance. Furthermore, H. Chen and T.J. Chen [19] investigated the strategic alliances between organizations by providing empirical evidence to show what kind of resources should be shared and how such resource sharing should be organized between partners. The partitioning of a set of objects is a process which partitions these objects into subsets, so that the objects of the same group have similar characteristics, and two objects belonging to two distinct subsets are dissimilar. In fact, this process is subjective because the same set of elements should often be divided differently for various applications. It is known that the partitioning process is involved in many applications occurred in various areas such as: clustering of data, load balancing in parallel machines, image segmentation, data mining, scientific computing, VLSI design, task scheduling, parallel programming, geographical information systems, division of space air, classification of documents, etc. One of the approaches used to solve the partitioning problem is to reduce it to a problem of graphs where the objects are represented by vertices and the edges correspond to possible relationships between the objects. Although some problems try to partition the edges of a graph, we usually mean by the partition of a graph the partition of the vertices of this graph. Once the graph is obtained, the problem is to find a partition of the graph into subgraphs according to a certain criterion. However, the main question is which criterion to choose so that we obtain the “best partition”? For example, the partition of a graph into the least number of independent sets is used to solve the chromatic number problem. On the other hand, there exist several studies on partitioning graphs into k-alliances and various theoretical results are obtained for some graphs classes in the literature. In this framework, the partitioning of graphs into defensive k-alliances is investigated by Eroh and Gera [20,21], Haynes and Lachniet [22], Yero et al. [23], the Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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partitioning into offensive k-alliances is the subject matter of Sigarreta et al. [24], Yero [25], and the partitioning into powerful k-alliances is discussed in Slimani and Kheddouci [26], Yero and Rodr´ıguez-Vel´azquez [27]. In this context, it would be interesting to see how such partitioning can be of interest to solve practical problems. Shafique [28] studied the partitioning of data (objects) into clusters by involving the concept of defensive 0-alliance. In general, the clusters are defined by maximizing the similarities of objects belonging to each cluster as well as the dissimilarities of objects between the clusters. Thus, there exist more bonds inside each cluster than between the clusters. He represented this situation by a graph where the vertices correspond to data and the edges symbolize the common property (similarity) that the data share. This implies that the vertices in every cluster have at least as many edges adjacent to the vertices inside the cluster as to the vertices outside it. This corresponds to the definition of defensive 0-alliance. On the basis of this, Shafique [28] established an approximate algorithm and applied it for different clustering applications. In mobile ad hoc networks (MANETs), ensure confidentiality and secure communications in groups is a critical task. In this context, Seba et al. [29] proposed a fully distributed and self-stabilizing clustering algorithm for key management in MANETs by using the concept of defensive (−1)-alliance as a clustering criterion. Thus, they proposed a solution which meets the criteria of self-organization and mobility resilience. With experiments, they showed that the concept of defensive (−1)-alliance is an efficient clustering criterion for group key management in MANETs by comparing to other existing clustering schemes. Recently, there have been two surveys in the literature on alliances: the first by Yero and Rodr´ıguez-Vel´azquez [3] presents essentially results on one type of k-alliances in graphs namely those are defensive, and in the second Fernau and Rodr´ıguez-Vel´azquez [30] have investigated the problem of several graph parameters which are known under completely different names in different areas and they have proposed a new framework called (global) (D, O)alliances in order to unify their notations. In this paper, we survey the vast mathematical properties of defensive, offensive and powerful k-alliances in graphs by presenting a large number of theoretical results corresponding to bounds and/or exact values obtained for the associated k-alliance (partition) numbers for various graph classes. There is an originality in our draft by surveying the most important results by classifying them firstly according to the “type of k-alliance” then by the criterion “graph class”. This allows us to see how the study of various k-alliance (partition) numbers varies according to the two criteria “graph class” and “type of k-alliance”. In particular, it is deduced the most/least studied graph classes (type of k-alliances) on which there are more/less results. The paper is partitioned into three principal parts: the first part given in Section 2 is devoted to the study of defensive k-alliances, the second part given in Section 3 deals with the offensive k-alliances, and the third part given in Section 4 discusses the powerful k-alliances. At the end of each part, we establish a table that summarizes the principal results obtained for every k-alliance (partition) number according to the different graph classes and we discuss some of their relationships and properties. Finally, in Section 5 we summarize and draw conclusions. 1.2. Terminology and definitions In this part, we give some terminology and definitions which will be heavily used in the rest of this paper. Let G = (V, E) be an undirected finite graph without loops and multiple edges where V denotes the vertex set and E denotes the edge set with |V | = n and |E| = m. For a non-empty subset S ⊆ V , ⟨S⟩ denotes the subgraph of G induced by S. For any vertex v ∈ V , N (v) is the open neighborhood of the vertex v, i.e. the set of vertices that are adjacent to v in G, and the closed neighborhood of v is the set N [v] = N (v) ∪ {v}. The number deg S (v) = |N S (v)| is the degree of v in S with N S (v) = {u⋃∈ S : uv ∈ E} is the set of neighbors v has in S and N S [v] = N S (v) ∪ {v}. The open neighborhood of S is N (S) = v∈S N (v) and the closed neighborhood of S is N [S] = N (S) ∪ S. The boundary of S is the set ∂ S = N (S) − S and the complement of S in V is S¯ = V − S. We denote the degree sequence of G by δ1 ≥ δ2 ≥ · · · ≥ δn . Other notations will be introduced when needed. A set S is a dominating set if N [S] = V and it is a total dominating set or an open dominating set if N (S) = V . The minimum cardinality of a dominating set (resp. total dominating set) of G is the domination number γ (G) (resp. total domination number γt (G)). ¯ A non-empty set of vertices S ⊆ V is called defensive alliance if for every vertex v ∈ S, |N [v] ∩ S| ≥ |N (v) ∩ S| or equivalently deg S (v) + 1 ≥ deg S¯ (v). In this case, we say that every vertex in S is defended from possible attack ¯ A defensive alliance S is called strong if for every vertex v ∈ S, deg S (v) + 1 > deg S¯ (v). A by vertices in S. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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¯ or equivalently set S ⊂ V is called offensive alliance, if for every vertex v ∈ ∂(S), |N (v) ∩ S| ≥ |N [v] ∩ S| deg S (v) ≥ deg S¯ (v) + 1. In this case, we say that every vertex in ∂(S) is vulnerable to possible attack by vertices in S. An offensive alliance S is called strong if for every vertex v ∈ ∂(S), deg S (v) > deg S¯ (v) + 1. The alliances that are both defensive and offensive are called powerful alliances. That is, S ⊂ V is a powerful alliance if for every vertex ¯ An alliance S of any type (defensive, offensive or powerful) is called global if S v ∈ N [S], |N [v] ∩ S| ≥ |N [v] ∩ S|. is a dominating set, and it is called critical or minimal if no proper subset of S is an alliance of the same type. See the graphs given in Figs. 1(a)–1(c) of Appendix for illustration. A subset S ⊆ V is a defensive k-alliance, with k ∈ {−δ1 , . . . , δ1 }, if for every v ∈ S, deg S (v) ≥ deg S¯ (v) + k. A set S ⊂ V is an offensive k-alliance, with k ∈ {2 − δ1 , . . . , δ1 } if for every v ∈ ∂(S), deg S (v) ≥ deg S¯ (v) + k. A set S ⊂ V is a powerful k-alliance if it is both a defensive k-alliance and an offensive (k + 2)-alliance. Yero and Rodr´ıguez-Vel´azquez [31,32] studied the limit case of defensive (resp. offensive and powerful) k-alliances and by considering equalities in their associated definitions they defined the so-called boundary defensive (resp. offensive and powerful) k-alliances. See the graphs given in Figs. 1(d)–1(f) of Appendix for illustration. 2. Defensive k-alliances in graphs In this section, we study mathematical properties of defensive k-alliances by giving bounds and/or exact values of several parameters studied for various graphs classes. A defensive k-alliance in a graph G = (V, E) is a set of vertices S ⊆ V satisfying the condition that every vertex in S has at least k more neighbors in S than it has outside of S. The case k = −1 (resp. k = 0) corresponds to the standard defensive alliances (resp. strong defensive alliances which is also known as a cohesive set) defined in [1]. Several parameters have been defined and studied in the literature for defensive k-alliances, one can see [1,2,33– 36] and others. These parameters are defined as follows: The defensive (−1)-alliance number known as defensive d alliance number a−1 (G) (resp. defensive 0-alliance number known as strong defensive alliance number a0d (G)) is the minimum cardinality among all (critical) defensive (−1)-alliances (resp. defensive 0-alliances) of G [1]. The global d defensive (−1)-alliance number γ−1 (G) (resp. global defensive 0-alliance number γ0d (G)) is the minimum cardinality among all (critical) global defensive (−1)-alliances (resp. global defensive 0-alliances) of G [2]. The upper defensive (−1)-alliance number Ad−1 (G) (resp. upper defensive 0-alliance number Ad0 (G)) is the maximum cardinality among all critical defensive (−1)-alliances (resp. defensive 0-alliances) of G [1,28]. The defensive k-alliance number akd (G) is the minimum cardinality among all (critical) defensive k-alliances of G [9,28]. The global defensive k-alliance number γkd (G) is the minimum cardinality among all (critical) global defensive k-alliances of G [37]. The upper defensive k-alliance number Adk (G) is the maximum cardinality among all critical defensive k-alliances of G [28]. Now, we give some basic relations and observations which bind various parameters of defensive k-alliances. For any graph G = (V, E), we have: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

d a−1 (G) ≤ a0d (G) ≤ Ad0 (G) [1,28]; d a−1 (G) ≤ Ad−1 (G) [1,28]; d d a−1 (G) ≤ γ−1 (G) [2,38]; a0d (G) ≤ γ0d (G) [28,38]; d a−1 (G) = 1 ⇔ ∃ v ∈ V, deg(v) ≤ 1 [1]; a0d (G) = 1 ⇔ G contains an isolated vertex [1]; d a−1 (G) = 2 ⇔ δn ≥ 2 and G has two adjacent vertices of degree at most 3 [1]; d a0 (G) = 2 ⇔ δn ≥ 1 and G has two adjacent vertices of degree at most 2 [1]; d a−1 (G) ≤ a0d (G) ≤ γ0d (G) [2,38]; d γ (G) ≤ γ−1 (G) ≤ γ0d (G) [2,38,39]; d akd (G) ≤ ak+1 (G) [23,34,37]; d ak (G) ≤ γkd (G) [23,37]; d γ (G) ≤ γkd (G) ≤ γk+1 (G) [23,37]; d d Ak (G) ≤ Ak+1 (G) [35].

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2.1. Study of defensive k-alliance numbers for various graphs classes Defensive k-alliances are extensively studied in the literature for different graphs classes. In this subsection, we present important theoretical results obtained for this type of alliance. Essentially, we give bounds or exact values established for defensive k-alliance numbers studied for various graphs classes. 2.1.1. General graphs We present essential results concerning defensive k-alliances in the case of general graphs. In particular, we give bounds obtained for various defensive k-alliance numbers by using different graph parameters. Let G = (V, E) be a general graph of order n. The study of defensive k-alliances in graphs was introduced by Kristiansen et al. [1]. They proposed some sharp bounds for the defensive (−1)-alliance number as follows: for any connected ⌈ ⌉ ⌈ ⌉ number and the defensive ⌊ ⌋0-alliance ⌊ ⌋ d graph G, a−1 (G) ≤ min{n − δ2n , n2 } and a0d (G) ≤ min{n − δ2n , n2 +1} (note that these bounds are attained, for example, for the complete graph G = K n ). Rodr´ıguez-Vel´azquez and Sigarreta [40] studied the relationship between the (global) defensive k-alliance numbers of a graph, its algebraic connectivity (the second smallest eigenvalue of the Laplacian matrix of the graph G) and its spectral radius (the largest eigenvalue of the adjacency matrix of the d graph G). They obtained lower bounds for the parameters a−1 (G) and a0d (G) by using δ⌉1 and ⌈ ⌈ the⌉maximum degree nµ d , a0d (G) ≥ n(µ+1) and the algebraic connectivity µ. Thus they showed that for a simple graph G, a−1 (G) ≥ n+µ n+µ ⌊ ⌋ ⌉ ⌈ δ1 n(µ− 2 ) for a simple connected graph G, a0d (G) ≥ (note that this latter bound is reached, for example, in the µ

following cases given in [40]: the complete graph, the Petersen graph, and the 3-cube graph). Other bounds for the same parameters are given by Araujo-Pardo and Barri`ere [41] by using the minimum degree of the graph and its girth. Haynes et al. [2] investigated the global defensive (−1)-alliance number and the global defensive √ 0-alliance d number. They obtained sharp bounds and showed that: if G is a graph of order n, then γ−1 (G) ≥ 4n+1−1 ⌈2 ⌉ and √ d γ0d (G) ≥ n; ⌊and⌋ for any graph G with no isolated vertices and minimum degree δn , γ−1 (G) ≤ n − δ2n and d (G) ≥ ⌈ r n⌉+1 . Moreover, Haynes γ0d (G) ≤ n − δ2n . They also obtained that if G is a graph of order n, then γ−1 2 et al. [2] obtained other results for the same parameters by using the total domination number γt (G). Thus, for d the global defensive (−1)-alliance number they showed that for any graph G with δn ≥ 2, γ−1 (G) ≥ γt (G), and d furthermore if δ1 ≤ 3 then γ−1 (G) = γt (G). For the global defensive 0-alliance number they showed that for any graph G with no isolated vertices, γ0d (G) ≥ γt (G). On the other hand, Rodr´ıguez-Vel´azquez and Sigarreta [40] gave lower bounds for these parameters in terms of the order of a simple graph G, its maximum degree ⌈ δ1 and⌉its spectral ⌈ ⌉ ⌈ n ⌉ d ⌈ n ⌉ 2n d d d radius λ. These results are: γ−1 (G) ≥ λ+2 , γ−1 (G) ≥ δ +3 , γ0 (G) ≥ λ+1 and γ0 (G) ≥ ⌊ δ1n⌋ . For more 1

2

+1

bounds on these parameters, one can see Favaron [39], Hsua et al. [42] and Sigarreta and Rodr´ıguez-Vel´azquez [36], where other concepts such as the minimum cardinality of an independent set, the dominating number, and the diameter of graph G are used. Yero et al. [23] presented some relations for the (global) defensive k-alliance number by considering the cases where the degrees of vertices and k are even/odd. Thus, they obtained that if every vertex of G has even degree and k is odd, k = 2l − 1, then every (global) defensive (2l − 1)-alliance in G is a (global) defensive (2l)-alliance d d and vice versa. Hence, in such a case, a2l−1 (G) = a2ld (G) and γ2l−1 (G) = γ2ld (G). Analogously, if every vertex of G has odd degree and k is even, k = 2l, then every defensive (2l)-alliance in G is a defensive (2l + 1)-alliance d d and vice versa. Hence, in such a case, a2ld (G) = a2l+1 (G) and γ2ld (G) = γ2l+1 (G). Rodr´ıguez-Vel´azquez et al. [34] and Sigarreta in his thesis [43] studied the defensive k-alliances and showed that for every k ∈ {−δn , ..., δ1 }, the ⌊ δn −k ⌋ ⌈ δn +k+2 ⌉ d ≤ a (G) ≤ n − , and if k ∈ {−δn , . . . , 0} one has the upper defensive k-alliance number satisfies k 2 2 ⌈ ⌉ bound akd (G) ≤ n+k+1 (note that these bounds are attained, for example, for the complete graph G = K n for 2 n every k ∈ {1 − n, . . . , n − 1}). Moreover, for every k, r ∈ Z such that −δn ≤ k ≤ δ1 and 0 ≤ r ≤ k+δ , they 2 d d showed that ak−2r (G) + r ≤ ak (G). They also gave other bounds by involving the algebraic ⌈ connectivity ⌉of G. Thus, for any connected graph G and k ∈ {−δn , . . . , δ1 },

akd (G)

≥

⌈ n(µ+k+1) ⌉ n+µ

and

akd (G)

≥

n(µ−

⌊

δ1 −k ) 2

µ

⌋

. By

using the isoperimetric number I(G) and the algebraic connectivity µ, Yero [25] and Yero et al. [23] obtained other bounds for the same parameter. They showed that for any graph G if it is partitionable into defensive k-alliances, then Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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⌉ akd (G) ≥ I(G) + k + 1 and akd (G) ≥ µ+2(k+1) . Note that this latter bound is reached for example for the graph 2 G = C3 × C3 for k = 0, given in [23,25], in this { ∑ case µ = } 3. We recall that the isoperimetric number of a graph |E(X,Y )| v∈S deg S¯ (v) G = (V, E) is defined as I(G) = min S⊂V :|S|≤ n2 or I(G) = min min{|X with X, Y ⊆ V . |S| |,|Y |} Fernau et al. [44] studied the global defensive k-alliances and established bounds for the global defensive ⌈ k-alliance ⌉ √ 4n+k 2 +k δn −k n d d ≤ γk (G) ≤ n − ⌈ 2 ⌉ and γk (G) ≥ ⌊ δ1 −k ⌋ number. Thus they presented that for any graph G, 2 2

+1

d γ−4 (G)

(note that these bounds are attained, for instance, for the graph given in Fig. 1(g) of Appendix where = 1). These results are also obtained by Rodr´ıguez-Vel´azquez et al. [37] and Sigarreta [43] by showing that these bounds d are a generalization of those obtained by Haynes et al. [2] for γ−1 (G) and γ0d (G) in the cases of general and bipartite graphs. Furthermore, Rodr´ıguez-Vel´azquez et al. [37] and Sigarreta [43] showed that for S a global defensive k-alliance of minimum cardinality in G, if W ⊂ S is a dominating set in G then for every r ∈ Z such that d d 0 ≤ r ≤ γkd (G) − |W |, γk−2r + ⌉ r ≤ γk (G). By using the spectral radius λ, Sigarreta [43] also obtained that ⌈ (G) n d for every graph G, γk (G) ≥ λ−k+1 . The upper defensive k-alliances have been studied by Sigarreta [35]. He established a bound for the upper defensive k-alliance number in⌈terms of⌉the order of G and its minimum degree. Thus for every k ∈ {−δn , . . . , δ1 } and for every graph G, Adk (G) ≤ 2n−δ2n +k ; and if every S ⊂ V such that |S| ≥ r is a defensive k-alliance, then Adk (G) ≤ r . He also gave an other upper bound by defining and using a new concept φkd (G) which is the largest cardinality of a maximal defensive k-alliance free set. A set X ⊂ V is defensive k-alliance free, if for all defensive k-alliance S, S \ X ̸= ∅ (X does not contain any defensive k-alliance as a subset). A defensive k-alliance free set X is maximal if for every v ̸∈ X , there exists S ⊆ X such that S ∪ {v} is a defensive k-alliance. Thus, for every k ∈ {−δ1 , . . . , δ1 } one has Adk (G) ≤ φkd (G) + 1. By considering the limit case of k-alliances, Yero in his thesis [25] and Yero and Rodr´ıguez-Vel´azquez [31] defined a new variant of k-alliances called boundary k-alliances. They studied mathematical properties of such alliances by obtaining in particular several bounds on the cardinality of k-alliance. Thus, if S is ⌉ every boundary ⌊ 2n−δdefensive ⌋ ⌈ n +k ≤ |S| ≤ (note that these two bounds a boundary defensive k-alliance in a graph G, then δn +k+2 2 2 are reached, for instance, for the complete graph G = K n for every k⌈∈ {1⌊ − n,⌋ .⌉. . , n − 1}). Furthermore, for a ⌈ ⌉ ⌋ ⌊ n(µ−

δ1 −k )

n(µ∗ − δn −k )

2 2 connected graph G, if S is a boundary defensive k-alliance in G then ≤ |S| ≤ and µ µ∗ ⌉ ⌈ ⌉ ⌈ n(µ+k+2)−µ ≤ |S| ≤ n − n(µ−k)−µ , where µ∗ is the Laplacian spectral radius (the largest Laplacian eigenvalue of 2n 2n the graph G).

2.1.2. Tree graphs By definition, a tree T = (V, E) of order n and size m is a connected graph with m = n − 1. The study of trees is particularly interesting because various applications are modeled and solved by using their properties. Thus, the study of alliances in general and defensive alliances in particular for this class of graph is important. In what follows, we present some results concerning defensive k-alliance numbers in trees. Kristiansen et al. [1] studied the defensive k-alliances and gave an exact value for the defensive (−1)-alliance d number and an upper bound for the defensive 0-alliance number. These results are: a−1 (T ) = 1 and a0d (T ) ≤ n. Haynes et al. [2] obtained upper bounds and sharp lower bounds for the global defensive (−1)-alliance number d , γ0d (T ) ≥ n+2 , and the global defensive 0-alliance number as follows: if T is a tree of order n, then γ−1 (T ) ≥ n+2 4 3 3n d γ−1 (T ) ≤ 5 for n ≥ 4 (with equality for the latter bound if and only if T ∈ T1 with T1 is a special family of trees [2]), and γ0d (T ) ≤ 3n for n ≥ 3 (with equality if and only if T belongs to a special family of trees T2 [2]). 4 Rodr´ıguez-Vel´azquez and Sigarreta [45] gave more general lower bounds for the same parameters by imposing a condition on the number of connected components of the subgraphs induced by the alliances. They showed that if S is a global defensive (−1)-alliance (resp. 0-alliance) in a tree T such that the subgraph ⟨S⟩ has c connected components, ⌉ (resp. |S| ≥ ⌈ n+2c ⌉). As a particular case of these results, if ⟨S⟩ is connected, they obtained lower then |S| ≥ ⌈ n+2c 4 3 d bounds for γ−1 (T ) and γ0d (T ) already proved by Haynes et al. in [2]. On the other hand, Chen and Chee [46] proved d that for a tree T of order n ≥ 3 having s support vertices, γ−1 (T ) ≤ n+s , with equality if and only if T belongs to 2 a special family of trees ξ [46] (we recall that a vertex of degree one is called a leaf and its neighbor is a support vertex). Bouzefrane et al. [47] showed that if T is a tree of order n ≥ 2 with l leaves and s support vertices, then Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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d γ−1 (T ) ≥ 3n−l−s+4 (with equality if and only if T = P2 or T ∈ T with T is a special family of trees [47]) and 8 d γ0 (T ) ≥ 3n−l−s+4 (with equality if and only if T belongs to a special family of trees F [47]). 6 Harutyunyan [48] studied the global defensive (−1)-alliance number for the complete t-ary tree Tt,d and for its particular case the complete binary tree Td = T2,d . A t-ary tree is a rooted tree where each vertex has at most t children. A complete t-ary tree is a t-ary tree in which all the leaves have the same depth and all the vertices except the leaves have t children; thus Tt,d is the complete t-ary tree with depth d. For a complete binary tree T⌈d of⌉order n, d d Harutyunyan gave an exact value for the global defensive (−1)-alliance number: γ−1 (Td ) = γ−1 (T2,d ) = 2n for any 5 d. For complete t-ary tree Tt,d , he obtained and upper bounds for the same by means of t and d. Thus, ⌊ ⌋ lower ⌊ parameter ⌋ d for d ≥ 2 and t ≥ 2, we have t d−1 t−1 + t d−1 + t d−2 ≤ γ−1 (Tt,d ) ≤ t d−1 t−1 + t d−1 + t d−2 + t d−3 . Moreover, 2 2 exact values for Tt,d , t = 3 and t = 4 are given in [48,49]. On the other hand, for any tree T of order n Harutyunyan d o presented a bound for γ−1 (T ) by using the global offensive (−1)-alliance number γ−1 (T ) (that we will study in the n d o next section). This result is γ−1 (T ) ≤ γ1 (T ) + 2 . By using the independence number β(T ) (the maximum cardinality of an independent set in T ), Chellali and Haynes [50] gave sharp bounds for the global defensive (−1)-alliance number d (T ) ≤ β(T ), furthermore they obtained that and the global defensive 0-alliance number as follows: for any tree T , γ−1 n+l−1 d for every nontrivial tree T with l leaves γ−1 (T ) ≤ 2 ; note that for l ≤ n5 this latter bound is an improvement of d the one (γ−1 (T ) ≤ 3n for n ≥ 4) given by Haynes et al. [2]. For the global defensive 0-alliance number they showed 5 that if T is a tree of order n ≥ 3 with s support vertices, then γ0d (T ) ≤ 3β(T2)−1 and γ0d (T ) ≤ β(T ) + s − 1. Favaron [39] compared the global defensive (−1)-alliance number and the global defensive 0-alliance number to d the independent domination number i. He obtained bounds in the forms i(T ) ≤ f (γ−1 (T )) and i(T ) ≤ g(γ0d (T )) for special families of trees, where f and g are functions. Rodr´ıguez-Vel´azquez and Sigarreta [37] and Sigarreta [43] considered global defensive k-alliances and they obtained a lower bound for the cardinality of every global defensive k-alliance in trees, by imposing a condition on the number of connected components of the subgraphs induced by the k-alliances. Thus, if⌉ S is a global defensive ⌈ k-alliance in T such that the subgraph ⟨S⟩ has c connected components, then |S| ≥ n+2c (note that the authors 3−k gave two unusual graphs for which this bound is reached). As a particular case of this result, if ⟨S⟩ is connected, d Rodr´ıguez-Vel´azquez and Sigarreta [37] and Sigarreta [43] obtained lower bounds for γ−1 (T ) and γ0d (T ) already proved by Haynes et ⌈al. in⌉[2]. Furthermore, as a consequence of this same result, they obtained that for every tree T . This latter bound is attained for k ∈ {−4, −3, −2, 0, 1} in the case of G = K 1,4 as given of order n, γkd (T ) ≥ n+2 3−k in [37,43].

2.1.3. Planar graphs We say that a graph is planar if one can draw it in the plan so that its edges do not cross. In this paragraph, we put on view the essential results obtained on defensive k-alliance parameters for this type of graphs. Let P = (V, E) be a planar graph of order n. A global alliance S is said to be an empire if with respect to a planar embedding of G, each connected component of ⟨S⟩ can be enclosed by a closed Jordan curve — a “wall” surrounding a fortress, where the region outside of each ¯ Enciso and Dutton [33] and Enciso [51] used this concept and showed that Jordan curve contains all vertices of S. for a planar graph P where S ⊆ V is a global defensive (−1)-alliance (resp. global defensive 0-alliance) of P, if S is an empire then |S| ≥ ⌈ n+6 ⌉ (resp. |S| ≥ ⌈ n+6 ⌉). Rodr´ıguez-Vel´azquez and Sigarreta [45] presented tight bounds in 6 5 planar graphs for the global defensive (−1)-alliance number and the global defensive 0-alliance number according to the order n as follows: ⌉ ⌈ d . (i) If n > 6, then γ−1 (P) ≥ n+12 8 ⌈ n+8 ⌉ d (ii) If n > 6 and P is a triangle-free graph, then γ (P) ≥ . −1 6 ⌈ n+12 ⌉ d (iii) If n > 4, then γ0 (P) ≥ . 7 ⌈ ⌉ (iv) If n > 4 and P is a triangle-free graph, then γ0d (P) ≥ n+8 . 5 Furthermore, they proved that if S is a global defensive (−1)-alliance in a general graph G such that the subgraph ⟨S⟩ n−2 f +4 is planar ⌈ connected ⌉ with f faces, then |S| ≥ ⌈ 4 ⌉ and in the case where S is a global defensive 0-alliance then n−2 f +4 |S| ≥ . Rodr´ıguez-Vel´azquez and Sigarreta [45] also showed that for a general graph G of order n where 3 S is a global defensive (−1)-alliance such that |S| ≥ 2, if ⟨S⟩ is planar and its minimum degree is at least σ , then Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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⌈ |S| ≥

)

–

9

√

σ −7+

⌈ ⌉ ⌉ √ (σ −7)2 +4(12+n) σ −5+ (σ −5)2 +4(8+n) ; moreover, if ⟨S⟩ is also a triangle-free graph, then |S| ≥ . Note 2 2

that, they presented several examples of graphs for which all the above bounds are reached. On the other hand, Rodr´ıguez-Vel´azquez and Sigarreta [37] and Sigarreta [43] presented tight lower bounds for the global defensive k-alliance number. They showed that for any planar graph P of order n: ⌈ ⌉ (i) If n > 2(2 − k), then γkd (P) ≥ n+12 . 7−k ⌈ ⌉ (ii) If n > 2(2 − k) and P is a triangle-free graph, then γkd (P) ≥ n+8 . 5−k They also obtained that if a simple general graph ⌈ G has⌉a global defensive k-alliance S such that the subgraph ⟨S⟩ f +4 is planar connected with f faces, then |S| ≥ n−2 . Note that, Rodr´ıguez-Vel´azquez and Sigarreta [37] and 3−k Sigarreta [43] presented examples of graphs where all these bounds are attained. Yero [25] and Yero and Rodr´ıguez-Vel´azquez [31] studied the boundary defensive k-alliances. They showed that if S is a boundary defensive k-alliance⌊in√a general graph G such ⌋ that ⟨S⟩ is planar connected with f faces then 16−8 f +(n+k−2)2 +n+k−2 C+4−2 f |S| = 2−k for k ̸= 2, and |S| ≤ , where C is the number of edges of G with one 2 endpoint in S and the other endpoint outside of S. Note that this latter bound is tight and it is attained for example for the complete graph G = K 5 where any subset S of G of cardinality four is a boundary defensive 2-alliance and ⟨S⟩ ∼ = K 4 as given in [25,31]. Furthermore, they presented lower and upper bounds for |S| according to the value of k. Thus, they proved that for a boundary defensive k-alliance S in a general graph G such that ⟨S⟩ is planar connected f −8 f −8 with f > 2 faces, if k ∈ {5 − δ1 , . . . , δ1 } (resp. k ∈ {5 − δn , . . . , δ1 }) then |S| ≥ ⌈ δ 4+k−4 ⌉ (resp. |S| ≤ ⌊ δn4+k−4 ⌋). 1

2.1.4. Complete graphs Let K n = (V, E) be a complete graph of order n. In this part, we exhibit some exact values obtained for defensive k-alliance numbers in complete graphs. Kristiansen et al. [1] investigated the defensive k-alliances and obtained ⌈ ⌉ exact values for ⌊ the ⌋ defensive (−1)-alliance d number and the defensive 0-alliance number as follows: a−1 (K n ) = n2 and a0d (K n ) = n2 + 1. Haynes et al. [2] studied the global defensive k-alliances and established exact values for the global⌊ defensive ⌋ d (K n ) = n+1 (−1)-alliance number and the global defensive 0-alliance number. These results are: γ−1 and 2 ⌉ ⌈ . γ0d (K n ) = n+1 2 Rodr´ıguez-Vel´azquez and Sigarreta [37], Rodr´ıguez-Vel´azquez et al. [34] and Sigarreta [43] considered the (global) defensive k-alliances and obtained an exact value for the⌈ (global) ⌉ defensive k-alliance number. Thus, they showed that for every k ∈ {1 − n, . . . , n − 1}, akd (K n ) = γkd (K n ) = n+k+1 . Sigarreta [35] ⌈studied⌉the upper defensive k-alliance 2 number and obtained the same value for this parameter as well, i.e. Adk (K n ) = n+k+1 . 2 Yero [25] and Yero and Rodr´ıguez-Vel´azquez [31] studied the boundary defensive k-alliances and proved that the cardinality of every boundary defensive k-alliance S in the complete graph is |S| = n+k+1 . 2 2.1.5. Bipartite graphs and complete bipartite graphs A graph is bipartite if its vertices can be divided into two sets X and Y so that every edge of the graph connects a vertex in X to a vertex in Y . Let B = (X, Y, E) be a bipartite graph of order n. K r,s is the complete bipartite graph where r (resp. s) is the cardinality of the set X (resp. Y ). Kristiansen et al. [1] studied the defensive k-alliances and established exact values for the defensive (−1)-alliance number and ⌊the⌋ defensive bipartite graphs. So they obtained that for 2 ≤ r ≤ s, ⌊ ⌋ 0-alliance number ⌈ ⌉ in ⌈ complete ⌉ d a−1 (K r,s ) = r2 + 2s and a0d (K r,s ) = r2 + 2s . Haynes et al. [2] investigated the global defensive k-alliances and obtained sharp lower bounds for the global d defensive (−1)-alliance number and the global defensive 0-alliance number in bipartite graphs γ−1 (B) ≥ δ 2n+3 and 1 γ0d (B) ≥ δ 2n+2 . Furthermore, they presented exact values for the same parameters in complete bipartite graphs as 1 ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌈ ⌉ ⌈ ⌉ d d follows: γ−1 (K 1,s ) = 2s + 1, γ−1 (K r,s ) = r2 + 2s if r, s ≥ 2, and γ0d (K r,s ) = r2 + 2s . Favaron [39] compared the global defensive (−1)-alliance number and the global defensive 0-alliance number to the independent domination d number i. He obtained bounds in the forms i(B) ≤ f (γ−1 (B)) and i(B) ≤ g(γ0d (B)) for special families of bipartite graphs, where f and g are functions. Sigarreta [35] studied the mathematical properties of upper defensive k-alliances in graphs and presented some results for the upper defensive k-alliance number in complete bipartite graph K r,s where r ≥ s. Thus, he proved Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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that: Adk (K r,s ) = 1, for⌋ k ∈ {−r, . . . , −s}; Adk (K r,s ) = ⌊ r −k d Ak (K r,s ) = r + s − 2 , for k ∈ {s, . . . , r − 1}.

⌈ r +k ⌉ 2

)

–

⌈ ⌉ + s+k , for k ∈ {−1, 0, . . . , s − 1}; and 2

2.1.6. Regular graphs A graph of which all the vertices have the same degree is known as regular and if the common degree is δ then it is said that the graph is δ-regular. In this paragraph, we give some bounds or exact values obtained for defensive k-alliance numbers concerning this class of graphs. For this, we denote by Rδ = (V, E) the δ-regular graph of order n. In the literature, the alliance numbers of δ-regular graphs are known only for small degrees. Kristiansen et al. [1] and Araujo-Pardo and Barri`ere [41] presented some exact values for the defensive (−1)-alliance number, the defensive 0-alliance number, the upper defensive (−1)-alliance number and the upper defensive 0-alliance number in δ-regular graphs. According to the value of δ and by using the concept of girth (the length of a smallest cycle in a graph) and lc(Rδ ) (the maximum length of an induced cycle in the graph), they obtained the results given as follows: (i) (ii) (iii) (iv) (v)

d a−1 (R1 ) = Ad−1 (R1 ) = 1 and a0d (R1 ) = Ad0 (R1 ) = 2. d a−1 (R2 ) = Ad−1 (R2 ) = a0d (R2 ) = Ad0 (R2 ) = 2. d a−1 (R3 ) = Ad−1 (R3 ) = 2, a0d (R3 ) = gir th(R3 ) and Ad0 (R3 ) = lc(R3 ). d a−1 (R4 ) = a0d (R4 ) = gir th(R4 ) and Ad−1 (R4 ) = Ad0 (R4 ) = lc(R4 ). d a−1 (R5 ) = gir th(R5 ) and Ad−1 (R5 ) = lc(R5 ).

Haynes et al. [2] studied the global defensive k-alliances and gave a lower bound for the global defensive (−1)d alliance number in 4-regular graph that is γ−1 (R4 ) ≥ n3 . Note that this bound is also true for cubic graphs as mentioned in [2]. Yero [25] and Yero and Rodr´ıguez-Vel´azquez [31] studied the boundary defensive k-alliances. They showed that for k ∈ {5 − δ, . . . , δ}, if S is a boundary defensive k-alliance in δ-regular graph such that ⟨S⟩ is planar connected 4 f −8 f −2) with f faces, then |S| = δ+k−4 , and C = 2(δ−k)( , where C is the number of edges of the graph with one endpoint δ+k−4 in S and the other endpoint outside of S. 2.1.7. Cycle graphs In this paragraph, we exhibit exact values obtained for defensive k-alliance numbers for this class of graphs. Let Cn = (V, E) be a cycle graph of order n. Kristiansen et al. [1] investigated the defensive k-alliances and showed that the different defensive k-alliance numbers, the defensive (−1)-alliance number, the defensive 0-alliance number, the upper defensive (−1)-alliance d number and the upper defensive 0-alliance number, have the same exact value which is equal to 2. Thus, a−1 (Cn ) = d d d a0 (Cn ) = A−1 (Cn ) = A0 (Cn ) = 2. Haynes et al. [2] studied the global defensive k-alliances and proved that in a cycle graph of order n ≥ 3 the global defensive (−1)-alliance number and the global defensive 0-alliance number are equal to the total domination number. d Thus, γ−1 (Cn ) = γ0d (Cn ) = γt (Cn ). 2.1.8. Path graphs Let Pn = (V, E) be a path graph of order n. Kristiansen et al. [1] showed that for any path graph Pn , the defensive d (−1)-alliance number satisfies a−1 (Pn ) = 1 and the defensive 0-alliance number verify a0d (Pn ) = 2. They also proved that for every path Pn with n ≥ 4 the upper defensive (−1)-alliance number and the upper defensive 0-alliance number are equal to the same value. Thus, Ad−1 (Pn ) = Ad0 (Pn ) = 2 for n ≥ 4. Haynes et al. [2] studied the global defensive k-alliances in paths and obtained some results for the global defensive (−1)-alliance number and the global defensive 0-alliance number. Thus, for any path Pn with n ≥ 3 they proved that the global defensive 0-alliance number is equal to the total domination number, i.e. γ0d (Pn ) = γt (Pn ). Furthermore, d for the global defensive (−1)-alliance number, they showed that: for n ≥ 2, γ−1 (Pn ) = γt (Pn ) unless n ≡ 2(mod 4), d in which case γ−1 (Pn ) = γt (Pn ) − 1. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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2.1.9. Line graphs A line graph L(G) of a graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges in G meet at one or both endpoints. In this part, we present theoretical results concerning defensive k-alliance parameters in line graphs. Let G = (V, E) be a graph of size m and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn . Let L(G) be the line graph of G. Sigarreta and Rodr´ıguez-Vel´azquez [36] studied mathematical properties of the defensive (−1)-alliance number, the defensive 0-alliance number, the global defensive (−1)-alliance number and the global defensive 0-alliance d number in line graphs. They obtained bounds for a−1 (L(G)) and a0d (L(G)) in terms of the maximum degree of δ +δ G (δ1 ), its ⌉minimum degree (δn ) and its second minimum degree (δn−1 ) as follows: ⌈ n 2n−1 ⌉ ≤ a0d (L(G)) ≤ δ1 , ⌈

δn +δn−1 −1 d ≤ a−1 (L(G)) ≤ δ1 (note that all these bounds are reached, for instance, in the case of G = C4 with 2 d d a−1 (L(C4 )) = a0 (L(C4 )) = 2). Moreover, if G has a unique vertex of maximum degree then the upper bound becomes d d a−1 (L(G)) ≤ δ1 − 1. They also showed that if G is a δ-regular graph with δ > 0 then a−1 (L(G)) = a0d (L(G)) = δ. d Furthermore, for a simple graph G, Sigarreta and Rodr´ıguez-Vel´azquez [36] gave bounds for γ−1 (L(G)) and⌈γ0d (L(G)) ⌉ 2m d , by means of the maximum degrees δ1 and δ2 of G and its size m. These bounds are: γ−1 (L(G)) ≥ δ +δ +1 1 2 ⌈ ⌉ ⌈√ ⌉ d γ0d (L(G)) ≥ δ 2m , and if m > 6 then γ−1 (L(G)) ≥ m+4−1 . 1 +δ2

Rodr´ıguez-Vel´azquez et al. [34] and Sigarreta [43] studied the defensive k-alliances and obtained bounds for the defensive k-alliance number in L(G). They showed that for every k ∈ {2 − δ1 − δ2 , . . . , δ1 + δ2 − 2}, akd (L(G)) ≥ ⌈ ⌉ δn +δn−1 +k (note that this bound is attained for instance for the graph L(K 4 ) for every k ∈ {2 − δ1 − δ2 , . . . , 0}, see 2 d Fig. 1(h) of Appendix ⌈ k ⌉in which a0 (L(K 4 )) = 3). Moreover, they proved that for every k ∈ {2(1 − δ1 ), . . . , 0}, d ak (L(G)) ≤ δ1 + 2 ; note that this upper bound is attained if G is a δ-regular graph [34,43]. Furthermore, Sigarreta [43] established an other lower bound by involving the algebraic µ of line graph L(G). He ⌈ connectivity ⌊ ⌋ ⌉l proved that the defensive k-alliance number is bounded by akd (L(G)) ≥

m(µl −

δ1 +δ2 −2−k ) 2

µl

.

Fernau et al. [44], Rodr´ıguez-Vel´azquez and Sigarreta [37] and Sigarreta in his thesis [43] presented a lower bound for the global defensive k-alliance number⌈ in L(G) by using the maximum degrees δ1 and δ2 of G and its size ⌉ m. Thus they obtained that γkd (L(G)) ≥

m

⌊

δ1 +δ2 −2−k +1 2

⌋

. Moreover, Sigarreta [43] established two other lower

d bounds for γk⌈ (L(G)). In fact, he obtained that for all graph G of size m and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn , ⌉ m ; Furthermore, if there exist in G two non adjacent vertices whose degrees are γkd (L(G)) ≥ √ (δ1 +δ2 −2)(δ1 +δ3 −2)−k+1 ⌉ ⌈ δ1 and δ2 then γkd (L(G)) ≥ δ +δ m−k−1 . 1

2

2.1.10. Cartesian product graphs Given two graphs G 1 = (V1 , E 1 ) and G 2 = (V2 , E 2 ) with the sets of vertices V1 = {v1 , v2 , . . . , vn 1 } and V2 = {u 1 , u 2 , . . . , u n 2 } respectively, the Cartesian product of G 1 and G 2 is the graph G 1 × G 2 = (V, E), where V = V1 × V2 and two vertices (vi , u j ) and (vk , u l ) are adjacent in G 1 × G 2 if and only if “vi = vk and (u j , u l ) ∈ E 2 ” ¯ i , i ∈ {1, 2}. or “(vi , vk ) ∈ E 1 and u j = u l ”. Let G i be a graph of order n i , minimum degree δ¯i and maximum degree ∆ Kristiansen et al. [1] studied the defensive k-alliances in Cartesian product graphs and obtained bounds d for the defensive (−1)-alliance number and the defensive 0-alliance number as follows: a−1 (G 1 × G 2 ) ≤ d d d d d d d min{a−1 (G 1 )a0 (G 2 ), a0 (G 1 )a−1 (G 2 )} and a0 (G 1 × G 2 ) ≤ a0 (G 1 )a0 (G 2 ). Chang et al. [49] obtained a lower bound for the global defensive (−1)-alliance⌈number⌉of a general graph. They d showed that if G is a graph of order n and maximum degree δ1 , then γ−1 (G) ≥

⌊ 2n ⌋ δ1 +3 2

. As a consequence they

established a lower bound for the global defensive (−1)-alliance of the Cartesian product of paths and cycles. ⌈ number ⌉ d Thus, if G i = Pni or Cni for i = 1, 2, then γ−1 (G 1 × G 2 ) ≥ n 13n 2 . Yero in his thesis [25] and Yero et al. [23] studied defensive k-alliances in Cartesian product graphs and gave some results for the defensive k-alliance number in G 1 × G 2 . They showed that if S1 is a defensive k1 -alliance in G 1 and S2 is a defensive k2 -alliance in G 2 , then S1 × S2 is a defensive (k1 + k2 )-alliance in G 1 × G 2 and akd1 +k2 (G 1 × G 2 ) ≤ akd1 (G 1 )akd2 (G 2 ); note that this bound is a general case of the results obtained by Kristiansen et al. [1]. They also Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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d ¯ 1, ∆ ¯ 2} ≤ s ≤ ∆ ¯1 +∆ ¯ 2 + k. obtained that ak−s (G 1 × G 2 ) ≤ min{akd (G 1 ), akd (G 2 )} where s ∈ Z such that max{∆ On the other hand, Yero [25] obtained that if G 1 × G 2 contains defensive k-alliances, then G i contains defensive d ¯ j )-alliances, with i, j ∈ {1, 2}, i ̸= j, and as a consequence a d (G 1 × G 2 ) ≥ max{a d (k − ∆ ¯ 2 (G 1 ) , ak−∆ ¯ 1 (G 2 )}. k k−∆ Yero [25] and Yero et al. [23] studied global defensive k-alliances in Cartesian product graphs and presented some bounds for the global defensive k-alliance number in G 1 × G 2 . Thus, they obtained that if G 1 contains a global ¯ 2 , . . . , δ¯2 }, G 1 × G 2 contains a global defensive (k1 + k2 )defensive k1 -alliance, then for every integer k2 ∈ {−∆ alliance and γkd1 +k2 (G 1 × G 2 ) ≤ γkd1 (G 1 )n 2 . And if G 2 contains a global defensive k2 -alliance, then for every integer ¯ 1 , . . . , δ¯1 }, G 1 × G 2 contains a global defensive (k1 + k2 )-alliance and γ d (G 1 × G 2 ) ≤ γ d (G 2 )n 1 . k1 ∈ {−∆ k1 +k2 k2

Remark 1. Let us note that: (i) The defensive k-alliances were studied in the literature for other graph classes such as star graphs, cubic graphs and circulant graphs. For more details, the reader can refer to [2,37,41–44]. (ii) Some results for the defensive k-alliance number akd (G) in the case of complement graphs are given by Sigarreta et al. [52]. Now, we summarize the results presented above by giving some bounds and exact values obtained for various parameters of defensive k-alliances for different graph classes. These results are given in Tables 1 and 2. Concluding remarks 1. As we can see from Tables 1 and 2, the defensive k-alliance numbers are studied for various graph classes. From this, we note that the most studied parameter is the global defensive (−1)-alliance number d (γ−1 (G)) and the least studied one is the upper defensive k-alliance number (Adk (G)). Furthermore, the general and tree graph classes are the most studied ones and the cycle and path graph classes are the least studied ones. Moreover, some parameters are not studied for certain graph classes. For example for the planar graphs class, several defensive d k-alliance numbers are not studied such as a−1 (P), akd (P) and Adk (P). Besides, for the regular graphs class, all the defensive k-alliance numbers with index k namely akd (Rδ ), γkd (Rδ ) and Adk (Rδ ) are not studied. However, the upper defensive k-alliance numbers are not studied for several graph classes. In particular, for Adk (G) just some results are given in the case of general and complete (bipartite) graph classes. Remark 2. The defensive (−1)-alliances as defined in the literature take into consideration the defense of a single vertex. In order to forestall any attack on the entire alliance or any subset of the alliance, Brigham et al. [53] proposed a model that take over this situation. Thus, they introduced the so-called secure sets as a generalization of the concept of defensive (−1)-alliances. Security and secure sets are studied in the literature and for more details one can refer to [53–55] and others. Remark 3. Rad and Rezazadeh [56] studied (strong) open alliances in graphs. According to their definition, an alliance is called open if it is defined completely in terms of open neighborhoods. They investigated the (strong) open defensive (resp. offensive) alliances by defining parameters called (strong) open defensive (resp. offensive) alliance number denoted by (aˆ t (G)) at (G) (resp. (aˆ ot (G)) aot (G)). Since at (G) ∼ = a0d (G) and aˆ ot (G) ∼ = a1o (G), Rad and Rezazadeh [56] established bounds only for aˆ t (G) and aot (G). 2.2. Study of defensive k-alliance partition numbers for some graph classes The partitioning of graphs into k-alliances is a process which partitions the set of vertices of a graph into subsets, so that each subset constitutes a k-alliance. The problem of partitioning a graph into defensive 0-alliances is introduced and studied by Gerber and Kobler [57] and Shafique and Dutton [58], and is referred to as “Satisfactory Graph Partitioning Problem (SGP)”. Thereafter, Shafique [28] investigated this partitioning problem and its application to data clustering. Moreover, Seba et al. [29] studied the partitioning of graphs into defensive (−1)-alliances and its application in mobile ad hoc networks (MANETs). Some parameters have been defined and studied in the literature for the partitioning into defensive k-alliances, these parameters are defined as follows: For any graph G = (V, E), the (global) defensive (−1)-alliance partition gd d number of G, (ψ−1 (G)) ψ−1 (G), is defined to be the maximum number of sets in a partition of V such that each set Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

d • ⌋) = ⌊ ra⌋−1 (K⌊r,s s 2 + 2 [1]

•B • K r,s

2

⌈n⌉

= 1 [1]

d (K n ) = • a−1

•

d a−1 (T )

[1] d • ⌈ ra⌉0 (K r,s ⌈ s)⌉= 2 + 2 [1]

2

⌊n⌋ + 1 [1]

[40]

≤ n [1]

• a0d (K n ) =

•

a0d (T )

n(µ− 21 ) µ

• a0d (G) ≥ ⌈ ⌊ ⌋ ⌉ δ

• a0d (G) ≥ ⌈ ⌉ n(µ+1) [40] n+µ ≥

≤

√ 4n+1−1 2⌈ ⌉[2] n − δ2n [2] ⌈ r n⌉ [2] 2 +1

d γ−1 (T ) d γ−1 (T )

1

[2]

2n [2] δ1 +3 ⌊ ⌋ d • γ−1 (K 1,s ) = 2s + 1 d •⌊ γ⌋−1 (K⌊r,s⌋) = r s 2 + 2 [2]

d • γ−1 (B) ≥

2

⌊ n+1 ⌋

(σ −7)2 +4(12+n) 2

d (K n ) = • γ−1

σ −7+

[45]

[2]

d • γ−1 (T ) ≤ β(T ) [50] d • γ−1 (T ) ≤ n+l−1 [50] 2 ⌈ ⌉ [33,51] • |S| ≥ n+6 6⌈ ⌉ d • γ−1 (P) ≥ n+12 8 ⌉ [45] ⌈ n−2 f +4 • |S| ≥ [45] 4 • |S| ≥ ⌉ ⌈ √

• ≥ n+2 4 [2] • ≤ 3n 5 [2] • |S| ≥ ⌈ n+2c 4 ⌉ [45] d • γ−1 (T ) ≤ n+s 2 [46] d • γ−1 (T ) ≥ 3n−l−s+4 [47] 8 d d • γ−1 (Td ) = γ−1 (T2,d ) = ⌈ 2n ⌉ 5 [48] ⌊ ⌋ • t d−1 t−1 + t d−1 + 2 d d−2 t ≤ γ−1 (Tt,d ) ≤ ⌊ ⌋ t d−1 t−1 + t d−1 + 2 t d−2 + t d−3 [48]

•

≥ γt (G) [2] ⌈ ⌉ n • ≥ λ+2 [40] ⌈ ⌉ 2n d • γ−1 (G) ≥ δ +3 [40]

•

•

d • γ−1 (G) ≥

• a0d (G) ≤ min{n − ⌊ ⌋ ⌊ ⌋ δn n 2 , 2 + 1} [1]

d • a−1 (G) ≤ ⌈ ⌉ ⌈ ⌉ min{n − δ2n , n2 }[1] ⌈ ⌉ nµ d • a−1 (G) ≥ n+µ [40] d γ−1 (G) d γ−1 (G) d γ−1 (G) d γ−1 (G)

d γ−1 (G)

a0d (G)

d a−1 (G)

Defensive alliance numbers

≥ [40]

⌈ n+1 ⌉

[2] 2 2n [2] δ1 +2 ⌈ ⌉ ⌈ ⌉ • γ0d (K r,s ) = r2 + 2s [2] • γ0d (B) ≥

• γ0d (K n ) =

⌈ ⌉ • |S| ≥ n+6 5⌈ [33,51] ⌉ n+12 d • γ0 (P) ≥ 7 ⌉ [45] ⌈ n−2 f +4 • |S| ≥ [45] 3

• γ0d (T ) ≥ n+2 3 [2] • γ0d (T ) ≤ 3n 4 [2] • |S| ≥ ⌈ n+2c 3 ⌉ [45] [47] • γ0d (T ) ≥ 3n−l−s+4 6 • γ0d (T ) ≤ 3β(T2 )−1 [50] • γ0d (T ) ≤ β(T )+s −1 [50]

⌊ n⌋ δ1 2 +1

≥ γt (G) [2] ⌈ ⌉ n λ+1 [40] ⌉ ⌈

√

n [2] ⌊ ⌋ ≤ n − δ2n [2]

• γ0d (G) ≥

•

•

•

γ0d (G) γ0d (G) γ0d (G)

• γ0d (G) ≥

γ0d (G) Ad−1 (G)

(continued on next page)

Ad0 (G)

)

Kn

P

T

G

Graph classes

Table 1 Previous results on defensive k-alliance numbers for various graph classes, with k ∈ {−1, 0}.

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• a0d (Pn ) = 2 [1] ⌈ ⌉ δ +δ • n 2n−1 ≤

d • a−1 (Pn ) = 1 [1] ⌈ ⌉ δn +δn−1 −1 • ≤ 2

Pn

G1× G2

L(G)

d • a−1 (G 1 × G 2 ) ≤ d min{a−1 (G 1 )a0d (G 2 ), d a0d (G 1 )a−1 (G 2 )} [1]

d a−1 (L(G))

• a0d (G 1 × G 2 ) ≤ a0d (G 1 )a0d (G 2 ) [1]

a0d (L(G))

≤ δ1 [36]

• a0d (Cn ) = 2 [1]

d • a−1 (Cn ) = 2 [1]

Cn

≤ δ1 [36]

• = 2, δ = 1 or 2 [1,41] • a0d (Rδ ) = gir th(Rδ ), δ = 3 or 4 [1,41]

• = δ, δ = 1 or 2 [1,41] d • a−1 (R3 ) = 2 [1,41] d • a−1 (Rδ ) = gir th(Rδ ), δ = 4 or 5 [1,41]

Rδ

a0d (Rδ )

a0d (G)

d a−1 (Rδ )

d a−1 (G)

Table 1 (continued) Graph Defensive alliance numbers classes

d • γ−1 (L(G)) ≥ ⌈√ ⌉ m + 4 − 1 [36] ⌈ ⌉ d • γ−1 (G 1 × G 2 ) ≥ n 13n 2 with G i = Pni or Cni for i = 1, 2, [49]

2

1

2

• γ0d (L(G)) ≥ ⌈ ⌉ 2m [36] δ +δ

d • γ−1 (L(G)) ≥ ⌈ ⌉ 2m δ +δ +1 [36] 1

• γ0d (Pn ) = γt (Pn ) [2]

d • γ−1 (Pn ) = γt (Pn ) [2]

[2]

γ0d (G)

• γ0d (Cn ) = γt (Cn ) [2]

≥

⌈ n3 ⌉

d • γ−1 (Cn ) = γt (Cn ) [2]

•

d γ−1 (R4 )

d γ−1 (G)

• Ad−1 (Pn ) = 2 [1]

• Ad−1 (Cn ) = 2 [1]

• = δ, δ = 1 or 2 [41] • Ad−1 (R3 ) = 2 [1,41] • Ad−1 (Rδ ) = lc(Rδ ), δ = 4 or 5 [1,41] Ad−1 (Rδ )

Ad−1 (G)

• Ad0 (Pn ) = 2 [1]

• Ad0 (Cn ) = 2 [1]

• Ad0 (Rδ ) = 2, δ = 1 or 2 [41] • Ad0 (Rδ ) = lc(Rδ ), δ = 3 or 4 [1,41]

Ad0 (G)

14 K. Ouazine et al. / AKCE International Journal of Graphs and Combinatorics ( ) –

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Table 2 Previous results on defensive k-alliance numbers for various graph classes. Graph classes

G

Defensive k-alliance numbers akd (G) ⌉ ⌈ ≤ akd (G) ≤ • δn +k+2 2 ⌊ ⌋ n − δn2−k [34,43] ⌈ ⌉ • akd (G) ≤ n+k+1 [34,43] 2 d • ak−2r (G) + r ≤ akd (G) [34,43] ⌉ ⌈ [34,43] • akd (G) ≥ n(µ+k+1) n+µ ⌈ ⌊ ⌋ ⌉ δ −k • akd (G) ≥

1 2

n(µ−

)

µ

[34,43]

γkd (G) √ 2 +k • 4n+k ≤ γkd (G) ≤ ⌈2 ⌉ n − δn2−k [37,43,44] ⌈ ⌉ • γkd (G) ≥

⌊

n⌋ +1

δ1 −k 2

Adk (G) • Adk (G) ≤ • •

Adk (G) Adk (G)

⌈

2n−δn +k 2

⌉

[35]

≤ r [35] ≤ φkd (G) + 1 [35]

[37,43,44]

d • γk−2r (G) + r ≤ γkd (G) [37,43] ⌉ ⌈ n [43] • γkd (G) ≥ λ−k+1

• akd (G) ≥ I(G) + k + 1 [23,25] ⌈ ⌉ [23,25] • akd (G) ≥ µ+2(k+1) 2 T

⌈

n+2c 3−k

• γkd (T ) ≥ P

Kn

• akd (K n ) =

⌈ n+k+1 ⌉ 2

[34,43]

⌉

[37,43] ⌉ [37,43] ⌈ ⌉ • γkd (P) ≥ n+12 [37,43] 7−k ⌈ ⌉ n−2 f +4 • |S| ≥ [37,43] 3−k ⌈ n+k+1 ⌉ d • γk (K n ) = [37,43] 2 • |S| ≥

⌈

n+2 3−k

• Adk (K n ) =

⌈ n+k+1 ⌉ 2

[35]

• Adk (K r,s ) = 1 ⌈ [35]⌉ ⌈ s+k ⌉ • Adk (K r,s ) = r +k + 2 [35] 2 ⌊ ⌋ • Adk (K r,s ) = r + s − r −k [35] 2

•B • K r,s Rδ Cn Pn L(G)

• akd (L(G)) ≥

δn +δn−1 +k [34,43] 2 ⌈k ⌉ δ1 + 2 [34,43]

⌈

• akd (L(G)) ≤ • akd (L(G)) ≥ ⌈ ⌊ ⌋ ⌉ δ +δ −2−k m(µl − 1 22 ) µl

G1 × G2

⌉

[43]

• akd1 +k2 (G 1 × G 2 ) ≤ akd1 (G 1 )akd2 (G 2 ) [23,25] d • ak−s (G 1 × G 2 ) ≤ min{akd (G 1 ), akd (G 2 )} [23,25] • akd (G 1 × G 2 ) ≥ d d max{ak− ¯ (G 1 ), ak−∆ ¯ (G 2 )} [25] ∆ 2

• γkd (L(G)) ≥ ⌈ ⌉ m

⌊

δ1 +δ2 −2−k 2

• γkd (L(G)) ⌈

⌋ +1

[37,43,44]

≥ ⌉

m √ [43] (δ1 +δ2 −2)(δ1 +δ3 −2)−k+1 ⌉ ⌈ m d • γk (L(G)) ≥ δ +δ −k−1 [43] 1 2

• γkd1 +k2 (G 1 × G 2 ) ≤ γkd1 (G 1 )n 2 [23,25] • γkd1 +k2 (G 1 × G 2 ) ≤ γkd2 (G 2 )n 1 [23,25]

1

of the partition is a (global) defensive (−1)-alliance [21,22]. The (global) defensive k-alliance partition number of gd G, (ψk (G)) ψkd (G), k ∈ {−δ1 , . . . , δn } is defined to be the maximum number of sets in a partition of V such that each set of the partition is a (global) defensive k-alliance [25]. We say that G is partitionable into (global) defensive gd k-alliances if (ψk (G) ≥ 2) ψkd (G) ≥ 2. Concerning the defensive k-alliance partition number, Yero et al. [23] have d given examples of extreme cases as follows: ψ−δ (G) = n where each set composed of one vertex is a defensive 1 d (−δ1 )-alliance, and ψδ (G) = 1 for the case of a connected δ-regular graph where V is the only defensive δ-alliance. In this subsection, we study mathematical properties of the (global) defensive (−1)-alliance partition number and the (global) defensive k-alliance partition number by presenting important theoretical results obtained for these

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parameters. Essentially, we give bounds or exact values for defensive k-alliance partition numbers studied for some graph classes. 2.2.1. General graphs Eroh and Gera [20] studied the basic properties of the defensive (−1)-alliance partition number by presenting general bounds by means of the minimum degree, the order and the girth⌊of graph G. For⌋ a connected graph G of √ d order n ≥ 3, they obtained sharp bounds given as follows: 1 ≤ ψ−1 (G) ≤ n + 23 − 1+4n . Furthermore, they gave 2 upper bounds by⌊involving ⌋ the minimum degree and the girth of G. Thus for a graph G having minimum degree δn , d then ψ−1 (G) ≤

⌈ n ⌉ δn +1 2

n d ⌋. , and if G is a graph with gir th(G) ≥ 3 and δn ≥ 4, then ψ−1 (G) ≤ ⌊ gir th(G)

On the other hand, Eroh and Gera [21] established an upper sharp bound for the global defensive (−1)-alliance ⌈ ⌉ gd partition number in a connected graph G having minimum degree δn . Thus, they proved that ψ−1 (G) ≤ 1 + δ2n . Yero [25] and Yero et al. [23] presented some relations for the (global) defensive k-alliance partition number by considering the cases where the degrees of vertices and k are even/odd. Thus, they obtained that if every vertex of G has even degree and k is odd, k = 2l − 1, then every (global) defensive (2l − 1)-alliance in G is a (global) defensive gd gd d (2l)-alliance and vice versa. Hence, in such a case, ψ2l−1 (G) = ψ2ld (G) and ψ2l−1 (G) = ψ2l (G). Analogously, if every vertex of G has odd degree and k is even, k = 2l, then every defensive (2l)-alliance in G is a defensive gd gd d (2l + 1)-alliance and vice versa. Hence, in such a case, ψ2ld (G) = ψ2l+1 (G) and ψ2l (G) = ψ2l+1 (G). Furthermore, d d they established a relation between the defensive k-alliance numbers ak (G) and ψk (G) by showing that their product is bounded by the order of graph, that is akd (G)ψkd (G) ≤ n. From this relation, they deduced that the lower bounds on d d akd (G) lead to upper bounds ⌉ ψk (G). For example, from the lower bound given for ak (G) by Rodr´ıguez-Vel´azquez ⌈ δn +k+2on d , they concluded that the defensive k-alliance partition number is bounded upperly by et al. [34],{ak (G) ≥ 2 ψkd (G) ≤

2n ⌋, δn + k + 2 2n ⌋, ⌊ δn + k + 3

⌊

δn + k even, δn + k odd.

Note that this latter bound for the even case is attained, for example, for the complete

d graph K 6 where ψ−3 (G) = 3 with the cardinality of each defensive (−3)-alliance is equal to 2. Furthermore, the corresponding bound for the odd case is reached, for instance, for the graph given in Fig. 1(i) of Appendix where d ψ−1 (G) = 2. Like the defensive k-alliance partition number, the global defensive k-alliance partition number is obtained from gd gd the relation between γkd (G) and ψk (G), and lower bounds of γkd (G). The relation between γkd (G) and ψk (G) given gd by Yero [25] and Yero et al. [23] is γkd (G)ψk (G) ≤ ⌈ n. By combining this relation and the lower bound obtained ⌉

by Rodr´ıguez-Vel´azquez and Sigarreta [37], γkd (G) ≥

n⌋

, Yero [25] and Yero et al. [23] obtained that the ⌋ ⌊ gd global defensive k-alliance partition number is bounded upperly by ψk (G) ≤ δ12−k + 1. They established other bounds for the global defensive k-alliance partition number. ⌋ Thus, they showed that for every graph G partitionable ⌊√ ⌊ ⌋ gd gd k 2 +4n−k and ψk (G) ≤ δn −k+2 . These latter bounds are attained, into global defensive k-alliances, ψk (G) ≤ 2 2 ⌊

gd

δ1 −k +1 2

gd

gd

for instance, in the following cases given in [23,25]: ψ−1 (K 4 × C4 ) = 4, ψ0 (K 3 × C4 ) = 3, ψ1 (K 2 × C4 ) = 2 gd and ψ1 (Ptr ) = 2, where Ptr denotes the Petersen graph. They also proved that for every k ∈ {1 − δn , . . . , δn } if gd gd ψk (G) ≥ 2, then γkd (G) + ψk (G) ≤ n+4 . By involving the algebraic connectivity µ, Yero [25] and Yero et al. [23] 2 ⌊ ⌋ gd showed that if any graph G is partitionable into global defensive k-alliances, then ψk (G) ≤ δ1 + 1 − µ2 − k . The authors in [23,25] gave an example of equality for this latter bound when the graph G = C3 × C3 for k = 0, in this case µ = 3. They obtained an other bound for the same { ∑ invariant } by using an other parameter of the graph G v∈S deg S¯ (v) which is the isoperimetric number I(G) = min S⊂V :|S|≤ n2 . Thus, they proved that for any graph G, if G |S| gd

is partitionable into global defensive k-alliances, then ψk (G) ≤ δ1 + 1 − I(G) − k. 2.2.2. Tree graphs Eroh and Gera [20] obtained upper and lower sharp bounds for the defensive (−1)-alliance partition number in d trees. Thus, they showed that for a tree T of order n ≥ 3, ψ−1 (T ) ≤ ⌊ 3n + 21 ⌋. Moreover, if T is a tree of order n ≥ 3 4 D d and diameter D ≥ 2 then ψ−1 (T ) ≥ ⌈ 2 ⌉ + 1. Furthermore, they proved that if T is a binary tree with a maximum d matching M (a matching is a subset M ⊂ E such that: u ∩ v = ∅ for each u, v ∈ M), then ψ−1 (T ) ≥ n − |M|. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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On the other hand, Eroh and Gera [21] showed that in a tree T of order n ≥ 3, the global defensive (−1)-alliance gd partition number is bounded by 1 ≤ ψ−1 (T ) ≤ 2. 2.2.3. Regular graphs Eroh and Gera [20] studied the defensive (−1)-alliance partition number in regular graphs and obtained ⌊ some ⌋ n ⌉ d upper bounds and an exact value for this parameter. Thus, for a δ-regular graph Rδ of order n, ψ−1 (Rδ ) ≤ ⌈ δ+1 , d if furthermore δ ≥ 3 and gir th(Rδ ) ≥ 5, then ψ−1 (Rδ ) ≤

2

n ⌉. ⌈ 1+(gir th(Rδ )−2) δ−3 2

As particular case, for a connected

d 3-regular graph having a maximum matching M, ψ−1 (R3 ) = |M|.

2.2.4. Cartesian product graphs Haynes and Lachniet [22] studied the⌊defensive partition number of grid graphs Pr × Pc and showed ⌋ ⌊ c−2(−1)-alliance ⌋ d that if 4 ≤ r ≤ c, then ψ−1 (Pr × Pc ) = r −2 + r + c − 2. 2 2 Yero [25] and Yero et al. [23] studied the defensive k-alliance partition number in Cartesian product graphs and they proved that for any graphs G 1 and G 2 , if there exists a partition of G i into defensive ki -alliances, i ∈ {1, 2}, then there exists a partition of G 1 × G 2 into defensive (k1 + k2 )-alliances and ψkd1 +k2 (G 1 × G 2 ) ≥ ψkd1 (G 1 )ψkd2 (G 2 ). ¯ i , i ∈ {1, 2}, they also showed that if s ∈ Z such that Moreover, for any graphs G i of order n i and maximum degree ∆ d ¯ ¯ ¯ ¯ max{∆1 , ∆2 } ≤ s ≤ ∆1 + ∆2 + k, then ψk−s (G 1 × G 2 ) ≥ max{n 2 ψkd (G 1 ), n 1 ψkd (G 2 )}. Furthermore, Yero [25] and Yero et al. [23] proved that if G i is partitioned into global defensive ki -alliances, gd i ∈ {1, 2}, then the global defensive k-alliance partition number of G 1 × G 2 is bounded by ψk1 +k2 (G 1 × G 2 ) ≥ gd gd max{ψk1 (G 1 ), ψk2 (G 2 )}. Moreover, they presented a relation between the global defensive (k1 + k2 )-alliance number of G 1 × G 2 and the global defensive ki -alliance partition number of G i , i ∈ {1, 2}. Thus, they obtained that for a gd n1 n2 . graph G i of order n i , i ∈ {1, 2}, if ψki (G) ≥ 1 then γkd1 +k2 (G 1 × G 2 ) ≤ gd maxi∈{1,2} {ψk (G i )} i

2.2.5. Partitioning a graph into boundary defensive k-alliances Yero [25] supposed G = (V, E) a graph and Πrd (G) = {S1 , S2 , . . . , Sr } a partition of V into r boundary defensive k-alliances and obtained tight bounds for r . Thus, he showed that if G can be partitioned into r boundary defensive 2n 2n 2n k-alliances, then 2n−δ ≤ r ≤ δn +k+2 (note that the complete graph K n can be partitioned into r = n+k+1 n +k boundary defensive k-alliances [25]). He also presented other tight bounds for r by using the algebraic connectivity 2µ ∗ µ and the Laplacian spectral radius µ∗ , these bounds are: 2µ∗2µ ≤ r ≤ 2µ−δ . An example where these −δn +k 1 +k bounds are reached is the complete graph G = K n as mentioned in [25]. Furthermore, he proved that for a graph ¯ where k ̸= δ1 G = (V, E) and C⌈⊂ E a⌉cut set partitioning V ⌊ ⌋ into two boundary defensive k-alliances S and S, 2m−kn 2m−kn 2m−kn (note that for a δ-regular graph |S| = n2 and and k ̸= δn , then 2(δ −k) ≤ |S| ≤ 2(δn −k) and |C| = 4 1

|C| =

n(δ−k) 4

as given in [25]). On the other hand, Yero [25] showed that if {X, Y ⌉} is a partition ⌈√ ⌊√of V into two boundary ⌋

n(kn−2m+nµ) n(kn−2m+nµ∗ ) defensive k-alliances in G, then without loss of generality, + n2 ≤ |X | ≤ + n2 and 4µ 4µ∗ √ √ ⌈ ⌉ ⌊ ⌋ n ∗) − n(kn−2m+nµ ≤ |Y | ≤ n2 − n(kn−2m+nµ) . 2 4µ∗ 4µ Now, we summarize the results presented above by giving some bounds and exact values obtained for defensive k-alliance partition numbers for some graph classes. These results are given in Table 3.

Concluding remarks 2. As we can see from Table 3, and comparing with Tables 1 and 2, we deduce that the defensive k-alliance partition numbers are studied on much less graph classes contrary to the defensive k-alliance numbers. For the studied graph classes (general, tree, regular, and Cartesian product graphs) the most studied parameter is the d defensive (−1)-alliance partition number (ψ−1 (G)) and the least studied one is the global defensive (−1)-alliance gd partition number (ψ−1 (G)). Furthermore, the general graph class is the most studied one and the regular graph class is the least studied one. Moreover, for the tree and regular graphs classes all the defensive k-alliance partition numbers gd with index k namely ψkd (G) and ψk (G) are not studied.

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G1× G2

Rδ

T

G

Graph classes

⌊

n gir th(G)

δn +1 2

⌈ n ⌉

⌋

[20]

[20]

[20]

n ⌈ ⌉ 1+(gir th(Rδ )−2) δ−3 2

δ+1 2

⌈ n ⌉

d • ψ−1 (P × P ) = ⌊ r −2 ⌋ ⌊ rc−2 ⌋ c + r + c − 2 [22] 2 2

d • ψ−1 (R3 ) = |M| [20]

d • ψ−1 (Rδ ) ≤

d • ψ−1 (Rδ ) ≤

⌊ ⌋ 1 d • ψ−1 (T ) ≤ 3n 4 ⌉+ 2 [20] ⌈ d • ψ−1 (T ) ≥ D 2 + 1 [20] d • ψ−1 (T ) ≥ n − |M| [20] ⌊ ⌋

d • ψ−1 (G) ≤

d • ψ−1 (G) ≤

d • 1 ≤ ψ−1 (G) ≤ ⌊ ⌋ √ 3 n + 2 − 1+4n [20] 2 ⌊ ⌋

d ψ−1 (G)

[20]

Defensive k-alliance partition numbers

gd

⌈

δn 2

⌉ [21]

• 1 ≤ ψ−1 (T ) ≤ 2 [21]

• ψ−1 (G) ≤ 1 +

gd

gd

ψ−1 (G)

Table 3 Previous results on defensive k-alliance partition numbers for some graph classes.

• ψkd1 +k2 (G 1 × G 2 ) ≥ ψkd1 (G 1 )ψkd2 (G 2 ) [23,25] d • ψk−s (G 1 × G 2 ) ≥ max{n 2 ψkd (G 1 ), n 1 ψkd (G 2 )} [23,25]

δn + k + 3

• akd (G)ψkd (G) ≤ n [23,25] d • ⎧ψk (G) ≤ ⎨⌊ 2n ⌋, δn + k even, δn + k + 2 [23,25] ⎩⌊ 2n ⌋, δn + k odd.

ψkd (G)

gd

gd

gd

K. Ouazine et al. / AKCE International Journal of Graphs and Combinatorics (

max{ψk1 (G 1 ), ψk2 (G 2 )} [23,25]

gd

• ψk1 +k2 (G 1 × G 2 ) ≥

• ψk (G) ≤ δ1 + 1 − I(G) − k [23,25]

gd

• γkd (G) + ψk (G) ≤ n+4 2 [23,25] ⌊ ⌋ gd • ψk (G) ≤ δ1 + 1 − µ2 − k [23,25]

gd

• γkd (G)ψk (G) ≤ n [23,25] ⌊ ⌋ gd • ψk (G) ≤ δ12−k + 1 [23,25] ⌊√ ⌋ gd k 2 +4n−k • ψk (G) ≤ [23,25] 2 ⌊ ⌋ gd • ψk (G) ≤ δn −k+2 [23,25] 2

gd

ψk (G)

18 ) –

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3. Offensive k-alliances in graphs In this section, we study mathematical properties of offensive k-alliances by giving bounds and/or exact values of several parameters studied for various graph classes. An offensive k-alliance in a graph G = (V, E) is a set of vertices S ⊂ V with the property that every vertex in the boundary of S has at least k more neighbors in S than it has outside of S. The case k = 1 (resp. k = 2) corresponds to the standard offensive alliances (resp. strong offensive alliances) defined in [1,4]. Several parameters have been defined and studied in the literature for offensive k-alliances, one can see [4,39,45,59–61] and others. These parameters are defined as follows: The offensive 1-alliance number known as offensive alliance number a1o (G) (resp. offensive 2-alliance number known as strong offensive alliance number a2o (G)) is the minimum cardinality among all (critical) offensive 1-alliances (resp. offensive 2-alliances) of G [1,4]. The global offensive 1-alliance number γ1o (G) (resp. global offensive 2-alliance number γ2o (G)) is the minimum cardinality among all (critical) global offensive 1-alliances (resp. global offensive 2-alliances) of G [5]. The upper offensive 1-alliance number Ao1 (G) (resp. upper offensive 2-alliance number Ao2 (G)) is the maximum cardinality among all critical offensive 1-alliances (resp. offensive 2-alliances) of G [1]. The offensive k-alliance number ako (G) is the minimum cardinality among all (critical) offensive k-alliances of G [8,9]. The global offensive k-alliance number γko (G) is the minimum cardinality among all (critical) global offensive k-alliances of G [44,59]. Now, we give some basic relations and observations which bind various invariants of offensive k-alliances for any graph G. (1) (2) (3) (4) (5) (6) (7) (8)

a1o (G) ≤ a2o (G) ≤ Ao2 (G) [1,28]; a1o (G) ≤ Ao1 (G) [1,28]; a1o (G) ≤ γ1o (G) [5]; a2o (G) ≤ γ2o (G) [5]; γ (G) ≤ γ1o (G) ≤ γ2o (G) [39]; o ako (G) ≤ ak+1 (G) [59]; o ak (G) ≤ γko (G) [25,43,59]; o γ (G) ≤ γko (G) ≤ γk+1 (G) [25,43,59].

3.1. Study of offensive k-alliance numbers for various graph classes Like defensive k-alliances the offensive k-alliances are studied in the literature for different graph classes. In this subsection, we present important theoretical results obtained for this type of alliance. We give bounds or exact values established for offensive k-alliance numbers studied for various graph classes. 3.1.1. General graphs In what follows, we present some theoretical results which exhibit various bounds for offensive k-alliance numbers in the case of general graphs. Let G = (V, E) be a general graph of order n and size m. Favaron et al. [4] explored the elementary properties of the offensive k-alliance numbers and they obtained bounds for the offensive 1-alliance number and the offensive 2-alliance number in general graphs. Thus they showed that: δn +1 ≤ a1o (G) ≤ γ (G)+n and a2o (G) > δn2+1 ; if every vertex of G has odd degree then a1o (G) ≤ n2 ; if n ≥ 2 then 2 2 a1o (G) ≤ 2n , and if n ≥ 3 then a2o (G) ≤ 5n . Moreover, they established that if δn ≥ 2 then this latter bound becomes 3 6 3n o a2 (G) ≤ 4 . Rodr´ıguez-Vel´azquez and Sigarreta [5] studied the global offensive k-alliances and presented several tight bounds for the global offensive 1-alliance number and the global offensive 2-alliance number in terms of several parameters of graph G. They showed that for all connected graph⌊G of order 1-alliance number ⌋ n ≥ 2, the ⌊ global offensive ⌋ ⌊ ⌋ o γ (G)+n n(2µ∗ −δn ) o o is bounded upperly by: γ1o (G) ≤ 2n , γ (G) ≤ , γ (G) ≤ and γ 1 1 1 (G) ≤ min{n − 3 2 2µ∗ ⌋ ⌊ n+β(G) β(G), }, where γ (G) (resp. µ∗ and β(G)) denotes the domination number (resp. Laplacian spectral radius 2 and independence number) of G. Note that, these bounds are attained, for instance, for the cocktail-party graph G = K6 − F ∼ = K 2,2,2 where n = µ∗ = 6, δn = 4, β(G) = γ (G) = 2 and γ1o (G) = 4 [5]. Moreover, they presented an other upper bound for γ1o (G) in the case of any connected graph G by means of its order and its maximum degree δ1 , Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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⌊

⌋ 1 that is γ1o (G) ≤ 2n−δ . On the other hand, Rodr´ıguez-Vel´azquez and Sigarreta [5] obtained tight upper bounds for 2 ⌊ ⌋ the global offensive 2-alliance number by proving that for all connected graph G of order n: γ2o (G) ≤ n+γ22 (G) , and ⌊ ⌋ (γ2 (G) denotes the 2-domination number of G which in addition if δn ≥ 2 then γ2o (G) ≤ n − β(G) and γ2o (G) ≤ 5n 6 is the minimum cardinality of a two dominating set; this latter is a dominating set where every vertex in S¯ is adjacent to at least two vertices in S). These previous results on global offensive k-alliances are also given by Sigarreta and Rodr´ıguez-Vel´azquez [60]. Note that to prove some of these results Rodr´ıguez-Vel´azquez and Sigarreta [5,60] used a new technique with respect to the one used by Favaron et al. [4] in their proof. o o Furthermore, Rodr´ıguez-Vel´azquez and Sigarreta [5,60] obtained ⌉ √ γ2 (G) in terms √ tight lower bounds for γ⌈1 (G) and of the order and the size of graph G, as follows: γ1o (G) ≥ ⌈ 3n−

9n 2 −8n−16m ⌉ 4

and γ2o (G) ≥

3n+1−

9n 2 −10n−16m+1 4

⌈

. ⌉

2m+n By involving the maximum degree of G, these bounds are improved by the same authors to obtain: γ1o (G) ≥ 3δ 1 +1 ⌈ ⌉ 2(m+n) o and γ2 (G) ≥ 3δ +2 (note that these two latter bounds are reached, for instance, in the case of the 3-cube graph 1 G = K 2 × K 2 × K 2 , where γ1o (G) = γ2o (G) = 4 as given in [5,60]). Moreover, by using the Laplacian spectral radius µ∗ and the minimum degree of G, Rodr´ azquez and Sigarreta ⌈ ⌈ıguez-Vel´ ⌉ presented other tight lower bounds ⌈ ⌈ [5,40,60] ⌉⌉ ⌉ δn +1 δn n n o o for the same parameters: γ1 (G) ≥ µ∗ and γ2 (G) ≥ µ∗ ( 2 + 1) . For these two latter bounds, if G is 2 the Petersen graph, then µ∗ = 5, γ1o (G) ≥ 4 and γ2o (G) ≥ 6 [5,40,60]. On the other hand, Rodr´ıguez-Vel´azquez and Sigarreta [40] gave other lower bounds for the same parameters. They showed that for a simple graph of order n, size m and maximum degree δ1 ,⌈the global√offensive 1-alliance ⌉ (2n+δ1 +1)− (2n+δ1 +1)2 −8(2m+n) number (resp. global offensive 2-alliance number) of G is bounded by γ1o (G) ≥ 4 ⌈ ⌉ √ 2 (2n+δ1 +2)− (2n+δ1 +2) −16(m+n) (resp. γ2o (G) ≥ ). Note that, this bound on γ1o (G) (resp. on γ2o (G)) is tight in the case 4

of the complete graph K n and the complete bipartite graph K 3,6 (resp. the complete bipartite graph K 3,3 ) [40]. In [60], Sigarreta and Rodr´ıguez-Vel´azquez studied the offensive k-alliances with connected subgraphs and showed ⌈ ⌉ ¯ is connected, |S| ≥ 3n−2 that for all minimal global offensive 1-alliance (resp. 2-alliance) S of G such that ⟨ S⟩ δ +3 1 ⌉ ⌈ ) (note that these bounds are attained, for example, for the cycle graph G = C , with (resp. |S| ≥ 4n−2 3 δ1 +4 γ1o (C3 ) = γ2o (C3 ) = 2). Other upper bounds for the global offensive 1-alliance number and the global offensive 2-alliance number are given by Harutyunyan [62]. Fernau et al. [59] and Sigarreta [43] studied the (global) offensive⌊ k-alliance ⌋ number and they showed that for n(2k+1) o any simple graph G and for all k ∈ {1, . . . , δn } one has γk (G) ≤ , and for any graph G and for every 2k+2 ⌈ δn +k ⌉ δn −k+2 o o k ∈ {2 − δn , . . . , δn } one has 2 ≤ ak (G) ≤ γk (G) ≤ n − ⌈ 2 ⌉ (note that these latter bounds are attained for every k in the case of the complete graph K n as mentioned in [43,59]). Furthermore, Fernau et al. [44,59] obtained lower and upper bounds for ⌈γko (G) by using the k-domination number γk (G) of a simple graph G and its Laplacian ⌊ ⌋ ⌈ ⌉⌉ spectral radius µ∗ , that are µn∗ δn2+k ≤ γko (G) ≤ γk (G)+n . On the other hand, Sigarreta [43] presented two 2 o lower bounds on γk (G) by means of ⌈ the order√of graph G, its size ⌉ and its maximum degree. These bounds are: ⌈ ⌉ (2n+δ1 +k)− (2n+δ1 +k)2 −8(2m+kn) 2m+kn o o γk (G) ≥ 3δ +k and γk (G) ≥ (note that these two bounds are attained, for 4 1

example, for the complete graph K n where γko (G) = 1 and k = 3−n). Moreover, Chellali et al. [63] obtained different bounds for γko (G) in terms of order, maximum degree, independence number, chromatic number and minimum degree. For instance, they proved that if G is a graph of order n with minimum degree δn ≥ k ≥ 1 (resp. δn ≥ k + 2 ≥ 4), k+1 k then γko (G) ≤ k+2 n, and this bound is best possible, (resp. γko (G) ≤ k+1 n). Also they showed that if G is a graph of o n +k) order n, minimum degree δn and maximum degree δ1 , then γk (G) ≥ 2δn(δ+δ . Besides, Volkmann [64] investigated 1 n +k o the connected global offensive k-alliance number γk,c (G) which is the minimum cardinality of a connected global o offensive k-alliance in the graph G. In this framework, he characterized connected graphs G with γk,c (G) = n(G) o o (γk,c (G) = n(G) − 1 in the case that δn ≥ k ≥ 2), and presented different tight bounds for γk,c (G). Yero and Rodr´ıguez-Vel´azquez [32] studied the mathematical properties ⌈ ⌉of boundary ⌊ powerful⌋ k-alliances and obtained that if S is a boundary offensive k-alliance in a graph G, then δn2+k ≤ |S| ≤ 2n−δn2+k−2 (note that these bounds are attained, for instance, for the complete graph G = K n for every k ∈ {3 − n, . . . , n − 1}).

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3.1.2. Tree graphs In this paragraph, we put on view some results concerning offensive k-alliance numbers in trees. Let T = (V, E) be a tree of order n. Favaron et al. [4] studied the offensive k-alliances and explored upper bounds for the offensive 1-alliance⌊number ⌋ and the offensive 2-alliance number in trees. Thus, they obtained that for any tree of n vertices, a1o (T ) ≤ n2 and ⌈ ⌉ a2o (T ) ≤ 3n . For the first bound, the equality is obtained for the path and the only other examples of equality are 4 K 1,3 with one edge subdivided once, and K 1,4 with two edges each subdivided once [4]. Rodr´ıguez-Vel´azquez and Sigarreta [45] studied the global k-alliances in planar graphs and presented some results for global offensive k-alliance numbers in trees. They obtained that if S is a global offensive 2-alliance in a tree such that the subgraph ⟨S⟩ has c connected components, then |S| ≥ n −c +1. Furthermore, they showed that if S⌈is a global ⌉ ¯ is a forest with c connected components then |S| ≥ 3(n−c)+1 offensive 1-alliance (resp. 2-alliance) in T such that ⟨ S⟩ 4 ⌉ ⌈ ). Moreover, Bouzefrane and Chellali [65] showed that for a tree T of order n ≥ 3 with l leaves (resp. |S| ≥ 4n−3c+1 5 and s support vertices, the global offensive 1-alliance number is bounded lowerly by γ1o (T ) ≥ n−l+s+1 (with equality 3 if and only if T belongs to a special family of trees F [65]). They also proved that if T ∈ F then γ1o (T ) = γ (T ). On the other hand, Favaron [39] compared the global offensive 1-alliance number and the global offensive 2-alliance number to the independent domination number i. He was interested in the existence of bounds in the forms γ1o (T ) ≤ f (i(T )) and i(T ) ≤ g(γ2o (T )) where f and g are functions. Thus, he obtained that for every tree T (resp. every tree T of order n ≥ 2), γ1o (T ) ≤ 2i(T ) − 1 (resp. i(T ) ≤ γ2o (T ) − 1), and these bounds are sharp. Harutyunyan [48] studied the global offensive k-alliances in complete t-ary trees and presented an exact value for the global offensive ⌊ n ⌋ 1-alliance number. Thus, they showed that for the complete t-ary tree Tt,d with depth d ≥ 1, γ1o (Tt,d ) = t+1 . On the other hand, Chellali [66] studied the offensive k-alliances in trees and proved that if k ≥ 2 and T belongs to a special family of trees Fk , then γko (T ) = γk (T ), with γk (T ) is the k-domination number of T . Moreover, Chellali and Volkmann [67] obtained an other exact value for the global offensive k-alliance number of any tree T of the family Fk by involving the cardinality of L σ (T ) which is the set of vertices having degree at most σ − 1. Thus, they showed that if T ∈ Fk , then γko (T ) = n+|L2σ (T )| . Furthermore, Sigarreta [43] presented a lower bound for the cardinality of every ¯ global offensive k-alliance. He proved that if S is a global offensive ⌈ ⌉ k-alliance in a tree T such that the subgraph ⟨ S⟩ n(k+2)−3c+1 is a forest with c connected components, then |S| ≥ (note that this bound is reached, for example, for k+3 the graph given in Fig. 1(j) of Appendix, where |S| = 2 and c = 4).

3.1.3. Planar graphs In this part, we present results concerning global offensive k-alliances in planar graphs. Let P = (V, E) be a planar graph of order n. Rodr´ıguez-Vel´azquez and Sigarreta [45] studied the global offensive k-alliances in planar graphs and showed that, for a planar graph P of order n > 2, if S is a⌈global offensive 1-alliance (resp. ⌉ ⌈ ⌉ 2-alliance) in P such that the subgraph ¯ has c connected components then |S| ≥ n−2c+4 (resp. |S| ≥ n−c+2 ). Moreover, for a planar graph P of order ⟨ S⟩ 3 2 ¯ is n, they proved that if S is a global offensive 1-alliance (resp. 2-alliance) in P such that the minimum degree of ⟨ S⟩ n(σ −1)+4 nσ +4 at least σ then |S| ≥ ⌈ σ +1 ⌉ (resp. |S| ≥ ⌈ σ +2 ⌉). Furthermore, Rodr´ıguez-Vel´azquez and Sigarreta [45] obtained other lower bounds for the cardinality of S which can be a global offensive 1-alliance or 2-alliance by using the ¯ Thus, they proved that, for a planar graph of order n, if S is a global number of faces of ⟨ S⟩. ⌈ ⌉offensive 1-alliance ⌈ ⌉(resp. ¯ is connected and has f faces, then |S| ≥ n+2 f (resp. |S| ≥ n+ f ). 2-alliance) in P such that the subgraph ⟨ S⟩ 3 2 Sigarreta [43] considered the global offensive k-alliances and obtained lower bounds concerning the cardinality of every global offensive k-alliance in planar graphs. Thus, he showed that if S is a global offensive k-alliance in a ¯ has c connected planar graph P of order n with k ∈ {1, 2, . . . , δ1 } (resp. k ∈ {0, 1,⌈. . . , δ1 }) such ⟨⌉S⟩ ⌉ that the subgraph ⌈ nk+2 f nk+2(2−c) ¯ (resp. |S| ≥ ) (note that these components (resp. ⟨ S⟩ is connected with f faces), then |S| ≥ k+2 k+2 bounds are attained, for example, for the graph given in Fig. 1(k) of Appendix where |S| = 3). Furthermore, he presented a lower bound for the global offensive k-alliance number, that is for a planar graph P of order ⌈ n and size⌉ m, o if P contains a global offensive k-alliance of minimum cardinality greater than two, then γk (P) ≥ 2m−n(6−k)+24 . 6+k Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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3.1.4. Complete graphs We exhibit in this paragraph some exact values obtained for offensive k-alliance numbers in complete graphs. Let K n = (V, E) be a complete graph of order n. Favaron et al. [4] studied the offensive k-alliances and established exact values for the offensive 1-alliance⌈number ⌉ and the offensive 2-alliance number in complete graphs. Thus, they obtained that for n ≥ 1, a1o (K n ) = n2 and ⌈ ⌉ a2o (K n ) = n+1 . Note that these values are examples of equality in bounds given in Section 3.1.1 obtained for 2 general graphs by Favaron et al. [4], that are a1o (G) ≥ δn2+1 and a2o (G) > δn2+1 . Fernau et al. [59] obtained an exact value for⌉the offensive k-alliance number in complete graphs, that is for ⌈ n+k−1 . It is clear, in this case, that every offensive k-alliance is global every k ∈ {3 − n, . . . , n − 1}, ako (K ) = n ⌈ n+k−1 ⌉ 2 and every vertex-set of cardinality is a global offensive k-alliance. Thus, as given by Bermudo et al. [68], 2 ⌈ ⌉ γko (K n ) = n+k−1 . 2 3.1.5. Bipartite graphs and complete bipartite graphs In this part, we present theoretical results representing bounds or exact values concerning offensive k-alliance numbers in bipartite graphs and complete bipartite graphs. Let B = (X, Y, E) be a bipartite graph of order n, and K r,s be a complete bipartite graph. Favaron et al. [4] studied the offensive k-alliances and presented some exact values for the offensive 1-alliance number and the offensive 2-alliance number in complete bipartite graphs. Thus, they obtained that: ⌈ ⌉ • for 1 ≤ r ≤ s, a1o (K r,s ) = ⌈ r +1 ;⌉ 2 ⌈ ⌉ • for 2 ≤ r ≤ s, a2o (K r,s ) = r2 + 1 , but a2o (K 1,s ) = 2s + 1 . Note that these values are examples of equality in bounds given in Section 3.1.1 obtained for general graphs by Favaron et al. [4], that are a1o (G) ≥ δn2+1 and a2o (G) > δn2+1 . Sigarreta and Rodr´ıguez-Vel´azquez [60] studied the global offensive k-alliances and established an upper bound for the global offensive 1-alliance number in bipartite graphs. Thus, they obtained that for all nontrivial bipartite graph, γ1o (B) ≤ n2 . Note that this⌊ bound ⌋ is an improvement of the one given by Rodr´ıguez-Vel´azquez and Sigarreta 2n o [5] for general graphs (γ1 (G) ≤ 3 ) in the case of bipartite graphs. Moreover, the same authors in [69] proved that this bound is an exact value for the global offensive 2-alliance number in the case of bipartite cubic graphs, that is γ2o (B) = n2 . On the other hand, Chellali [70] obtained other bounds for γ1o (B) and γ2o (B). Thus, he showed that for every bipartite graph B without isolated vertices, having l vertices of degree one (and s support vertices), γ2o (B) ≤ n+l 2 (and γ1o (B) ≤ n−l+s ). 2 By using L σ (B) (the set of vertices having degree at most σ − 1), Chellali and Volkmann [67] established a bound for the global offensive k-alliance number in bipartite graphs. They proved that for an integer σ ≥ 1, one has γko (B) ≤ n+|L2σ (B)| . On the other hand, Bermudo et al. [68] and Yero [25] gave bounds for the same parameter in complete bipartite graphs. Thus, they showed that for a complete bipartite graph K r,s with s ≤ r and for every k ∈ {2 − r, . . . , r } : (i) (ii) (iii) (iv)

If k ≥ s + 1,⌈then⌉γko (K r,s ) ⌉= r . ⌈ s+k If k ≤ s and r +k + s, ⌉then γko (K r,s ) = s. 2 ⌈ r +k ⌉2 ⌈≥s+k ⌈ ⌉ ⌈ s+k ⌉ If −s < k ≤ s ⌈and ⌉ 2 ⌈ + ⌉ 2 < s, then γko (K r,s ) = r +k + 2 ⌈ r +k 2⌉ . s+k o If k ≤ −s and r +k + < s, then γ (K ) = min{s, 1 + }. r,s k 2 2 2

3.1.6. Regular graphs We present in this paragraph some results obtained for offensive k-alliance numbers in regular graphs. We denote by Rδ = (V, E) the δ-regular graph of order n. Rodr´ıguez-Vel´azquez and Sigarreta [69] studied mathematical properties of the global offensive k-alliance numbers of cubic graphs and presented lower and upper bounds for the global offensive 1-alliance number in δ-regular graphs. Thus, they showed that for all δ-regular graph Rδ of order n and odd degree δ, n(δ+1) ≤ γ1o (Rδ ) ≤ n2 . In the case 3δ+1 of regular graphs of odd degree, this upper⌊ bound ⌋ is an improvement of the one given by Rodr´ıguez-Vel´azquez and Sigarreta [5] for general graphs (γ1o (G) ≤ 2n ). The same authors in [60] established an upper bound for the global 3 ⌊ ⌋ offensive 2-alliance number in 3-regular connected graph, that is γ2o (R3 ) ≤ 3n . 4 Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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Bermudo et al. [68] and Yero [25] investigated the relationships between global offensive k-alliances and some characteristic sets of a graph including r -dependent sets. They obtained an exact value for the offensive k-alliance number in δ-regular graphs, with δ > 0, by using a parameter of graphs, which is the maximum cardinality of an r -dependent set αr (Rδ ) (for a graph G = (V, E), a set S ⊆ V is an r -dependent set in G if the maximum degree of every vertex in the subgraph ⟨S⟩ induced by S is at most r i.e. deg S (v) ≤ r , ∀ v ∈ S). Thus, they showed that for every k ∈ {1, . . . , δ}, γko (Rδ ) = n − α⌊ δ−k ⌋ (Rδ ). 2

3.1.7. Cycle graphs Let Cn = (V, E) be a cycle graph of order n. In this part, we exhibit some results obtained for offensive k-alliance numbers in this class of graphs. Favaron et al. [4] studied the offensive k-alliances and they obtained that the offensive 1-alliance number⌈ and ⌉ the offensive 2-alliance number have the same value. Thus, they established that for n ≥ 3, a1o (Cn ) = a2o (Cn ) = n2 . The problem of finding the global offensive k-alliance number is N P-complete. Even so, for some graphs it is possible to obtain this number [25,68]. For instance, the family of the complete graphs, the cycle graphs and the path graphs. Thus, Bermudo et al. [68] and Yero [25] obtained exact values for the global offensive k-alliance number in {n ⌈ ⌉ for k = 0, cycle graphs for small specific values of k, in terms only of the order of graph, that is γko (Cn ) = n3 ⌈ ⌉ 2

for k = 1, 2.

3.1.8. Path graphs Let Pn = (V, E) be a path graph of order n. Favaron et al. [4] studied the offensive k-alliances and established exact values for the ⌊ ⌋offensive 1-alliance ⌊ ⌋ number and the offensive 2-alliance number. Thus they obtained that for n ≥ 1, a1o (Pn ) = n2 and a2o (Pn ) = n2 + 1. Bermudo et al. [68] and Yero [25] studied the global offensive k-alliances and presented exact values for the global offensive k-alliance number in path graphs for small specific values of k. Thus, they obtained that γko (Pn ) = { n ⌈ ⌉ 3 n ⌊ ⌋+k−1 2

for k = 0,

for k = 1, 2.

3.1.9. Line graphs In this part, we exhibit some results obtained for offensive k-alliance numbers in line graphs. Let G = (V, E) be a graph and L(G) its associated line graph. Sigarreta [43] studied the offensive k-alliances and obtained a lower bound for the global offensive ⌈ k-alliance ⌉ number in the line graph L(G). Thus, from a bound obtained for general graphs (that is γko (G) ≥ 2m+kn ), he ⌉ ⌈ ∑ 3δ1 +k n δ 2 +m(k−2) i=1 i . deduced that for a graph G of size m having a degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn , γko (L(G)) ≥ 3(δ +δ −2)+k 1

2

Furthermore, Fernau et al. [59] and Sigarreta [43] presented a lower bound for γko (L(G)) where G is a⌈δ-regular⌉graph. Thus, they showed that if L(G) is a line graph of a δ-regular graph G of order n, then γko (L(G)) ≥ n4 2(δ−1)+k (note 2 that this bound is attained, for instance, for the graph given in Fig. 1(l) of Appendix, with γ1o (L(G)) = γ2o (L(G)) = 3). On the other hand, Sigarreta [43] deduced that if G is a cubic graph of order n then 3n ≤ γ2o (L(G)) = γ1o (L(G)) ≤ n. 4 Note that these latter bounds are tight. For example, Sigarreta [43] mentioned that the upper one is reached in the case of the complete graph K 4 : γ1o (L(K 4 )) = 4 = n. Moreover, in the case of the complete bipartite graph K 3,3 , he obtained that γ1o (L(K 3,3 )) = 5 and for the lower bound 29 ≤ γ1o (L(K 3,3 )). 3.1.10. Cartesian product graphs ¯ i , i ∈ {1, 2}. Let G i = (Vi , E i ) be a graph of order n i , minimum degree δ¯i and maximum degree ∆ Yero and Rodr´ıguez-Vel´azquez [61] obtained various closed formulas for the global offensive 1-alliance number of several families of Cartesian product graphs, given as follows: • For any graphs G 1 and G 2 , γ1o (G 1 × G 2 ) ≥ 21 max{γ (G 1 )γ1o (G 2 ), γ1o (G 1 )γ (G 2 )}. Moreover, if G 1 has an efficient dominating set (S is an efficient dominating set if each vertex in S¯ is adjacent to exactly one vertex in S), then γ1o (G 1 × G 2 ) ≥ γ (G 1 )γ1o (G 2 ). Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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⌈ ⌉ (n−1)γ1o (G) • Let Pn be a path graph of order n. For every graph G of minimum degree δ¯ ≥ 1, γ1o (G × Pn ) ≥ + 2 ⌈ ⌉ δ¯ . 2 ⌈ o ⌉ nγ1 (G) . • Let Cn be a cycle graph of order n. For every graph G, γ1o (G × Cn ) ≥ 2 n1 n2 o • If Bi is a connected bipartite graph of order n i , i ∈ {1, 2}, then γ1 (B1 × B2 ) ≤ 2 . ⌊ ⌋ • The global offensive 1-alliance number of bamboo graph K r × Pt is γ1o (K r × Pt ) = r2t . • For any complete graph K r and any path graph Pt , γ1o (K r × Pt ) ≥ γ1o (K r )γ1o (Pt ). • If G 1 is a graph partitionable into two global offensive 1-alliances X 1 and X 2 and G 2 is a graph partitionable into two global offensive 2-alliances Y1 and Y2 , then γ1o (G 1 × G 2 ) ≤ |X 1 ∥Y1 | + |X 2 ∥Y2 |. • For any torus graph Cr × Ct , γ1o (Cr × Ct ) ≥ γ1o (Cr )γ1o (Ct ). ⌈ ⌉ • The global offensive 1-alliance number of the graph K r × Ct is γ1o (K r × Ct ) = r2t . • For any path graph Pr and any cycle graph Ct , γ1o (Pr × Ct ) ≥ γ1o (Pr )γ1o (Ct ). ⌈ ⌉ • Let r and t be two positive integers. If r⌉, t have the same parity, then γ1o (K r × K t ) = r2t , and if r ⌈ ⌈ ⌉ +t−1) ≤ γ1o (K r × K t ) ≤ r2t . Moreover, for any complete graphs and t have different parity then r t(r 2(r +t) γ1o (K r × K t ) ≥ γ1o (K r )γ1o (K t ). • Let Pr × Pt be a grid graph. ▷ If r and t are even, then γ1o (Pr × Pt ) = r2t . ⌈ ⌉ ▷ If r is even and t is odd, then γ1o (Pr⌈×⌉Pt ) ⌈= ⌉r (t−1) + r3 . 2 ▷ If r and t are odd, then (r −1)(t−1) + r3 + 3t ≤ γ1o (Pr × Pt ) ≤ 2

r (t−1) 2

+

⌈r ⌉ 3

.

Yero in his thesis [25] obtained bounds for the offensive k-alliance number in Cartesian product graphs. He showed that: ¯ 1 , k1 − ∆ ¯ 2 }, S1 × S2 is an (i) If Si is an offensive ki -alliance in G i = (Vi , E i ), i ∈ {1, 2}, then, for k = min{k2 − ∆ offensive k-alliance in G 1 × G 2 . (ii) If S1 × S2 is an offensive k-alliance in G 1 × G 2 , with Si ⊂ Vi , i ∈ {1, 2}, then S1 is an offensive (k + δ¯2 )-alliance ¯ 1 − δ¯2 , ∆ ¯ 2 − δ¯1 }. in G 1 and S2 is an offensive (k + δ¯1 )-alliance in G 2 , moreover, k ≤ min{∆ ¯ 2 , k2 − ∆ ¯ 1 }, a o (G 1 × G 2 ) ≤ a o (G 1 )a o (G 2 ). Note that As a consequence, he obtained that for every k ≤ min{k1 − ∆ k k1 k2 o there is equality for the graph C4 × K 4 , that is a−3 (C4 × K 4 ) = 2 = a0o (C4 )a1o (K 4 ) [25]. On the other hand, Yero [25] studied the global offensive k-alliance number in Cartesian product graphs and he showed that: ¯ 2 )-alliance in G 1 × G 2 . (a) If S is a global offensive k-alliance in G 1 , then S × V2 is a global offensive (k − ∆ (b) If S × V2 is a global offensive k-alliance in G 1 × G 2 , then S is a global offensive (k + δ¯2 )-alliance in G 1 , ¯ 1 − δ¯2 . moreover, k ≤ ∆ ¯ 2, As a consequence, he obtained that for any graph G 1 and any graph G 2 of order n 2 and maximum degree ∆ o o γk− (G × G ) ≤ n γ (G ). Furthermore, he established that the result given in (a) above can be simplified in 1 2 2 k 1 ¯2 ∆ the case of G 2 is a regular graph. In fact, for G 2 = (V2 , E 2 ) a δ-regular graph, a set S is a global offensive k-alliance in G 1 if and only if S × V2 is a global offensive (k − δ)-alliance in G 1 × G 2 . Remark 4. Let us note that the offensive k-alliances were studied in the literature for other graph classes such as star graphs and cubic graphs [39,61,69]. Now, we summarize the results presented above by giving some bounds and exact values obtained for various offensive k-alliance numbers for different graph classes. These results are given in Table 4: Concluding remarks 3. As we can see from Table 4, the most studied parameter is the global offensive k-alliance number (γko (G)) and the least studied one is the offensive k-alliance number (ako (G)). Furthermore, the general and tree graph classes are the most studied ones and the line graphs class is the least studied one. Moreover, some parameters are not studied for all or certain graph classes. For example, the upper offensive 1-alliance number Ao1 (G) and the upper offensive 2-alliance number Ao2 (G) are not studied for all graph classes. Besides, for the line graphs class, only Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

• a1o (K r,s ) =

•B • K r,s

=

2

⌈ r +1 ⌉

2

[4]

[4]

⌈n⌉

•

a1o (K n )

2

⌊n⌋

[4]

4

⌈ 3n ⌉

• a2o (K ⌈ ⌉ n) = n+1 [4] 2 o (K • a ⌈ r 2 r,s ⌉ )= 2 + 1 [4] • a2o (K 1,s ⌈ ⌉ )= s 2 + 1 [4]

• a2o (T ) ≤ [4]

δn +1 2 [4] 5n 6 [4] 3n 4 [4]

δn +1 2

• γ1o (B) ≤ • γ1o (B) ≤

• |S| ≥

n 2 [60] n−l+s [70] 2

⌈ n−2c+4 ⌉ [45] ⌈ 3 ⌉ • |S| ≥ n(σσ−1)+4 [45] +1 ⌉ ⌈ n+2 f [45] • |S| ≥ 3

•

• • •

3n−2 δ1 +3

γ1o (T ) ≥ n−l+s+1 [65] 3 γ1o (T ) = γ (T ) [65] γ1o (T ) ≤ 2i(T ⌊ ) −⌋1 [39] n γ1o (Tt,d ) = t+1 [48]

⌈

[5,40,60] ⌉ [60] • |S| ≥ ⌈ ⌉ • |S| ≥ 3(n−c)+1 [45] 4

n µ∗

⌈ ⌉ 2m+n • γ1o (G) ≥ 3δ [5,60] 1 +1 o • γ (G) ≥ ⌈ 1⌈ ⌉⌉

n(2µ∗ −δn ) [5,60] 2µ∗ • γ1o (G) ≤ min{n ⌊ ⌋ − β(G), n+β(G) } [5,60] 2 ⌋ ⌊ 2n−δ1 o • γ1 (G) ≤ [5,60] 2 o • γ (G) ≥ ⌈ 1 √ ⌉ 3n− 9n 2 −8n−16m [5,60] 4

o • ⌋ ⌊ γ2 (G) ≤

⌊ ⌋ • γ1o (G) ≤ 2n [5,60] ⌋ ⌊3 • γ1o (G) ≤ γ (G)+n [5,60] 2 o • γ (G) ≤ 1 ⌊ ⌋

⌈ 2(m+n) 3δ1 +2

⌉ [5,60]

• γ2o (B) = • γ2o (B) ≤

n 2 [69] n+l 2 [70]

⌈ n−c+2 ⌉ [45] ⌈ 2 ⌉ +4 [45] • |S| ≥ nσ ⌈ σ +2⌉ • |S| ≥ n+2 f [45] • |S| ≥

• |S| ≥ ⌈ n − c + 1⌉[45] [45] • |S| ≥ 4n−3c+1 5 • i(T ) ≤ γ2o (T ) − 1 [39]

1

⌉ + 1) [5,40,60] ⌈ ⌉ • |S| ≥ 4n−2 δ +4 [60]

δn n µ∗ ( 2

• γ2o (G) ≥ •⌈ γ2o (G) ⌈ ≥ ⌉

•

ako (K n )

=

2

⌈ n+k−1 ⌉

[59]

ako (G) ⌈ ⌉ • δn2+k ≤ ako (G) ≤ ⌈ ⌉ n − δn −k+2 [43,59] 2

n+γ2 (G) [5,60] 2 • γ2o (G) ≤ n ⌊ − ⌋β(G) [5,60] • γ2o (G) ≤ 5n 6 [5,60] •⌈ γ2o (G) ≥ ⌉ √ 3n+1− 9n 2 −10n−16m+1 [5,60] 4

γ2o (G)

γ1o (G)

⌉

[43]

–

Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

(continued on next page)

• γko (K n⌉) = ⌈ n+k−1 [59,68] 2 • γko (B) ≤ n+|L2σ (B)| [67] • γko (K r,s ) = r [25,68] • γko (K r,s ) = s [25,68] o = ⌉ • k (K ⌈ rγ+k ⌉ r,s ⌈) s+k + 2 [25,68] 2 • γko (K r,s ) = ⌈ ⌉ min{s, 1 + r +k 2 } [25,68]

2m−n(6−k)+24 6+k

⌉

nk+2(2−c) [43] ⌈ k+2⌉ nk+2 f [43] k+2

⌈

• |S| ≥ • γko (P) ≥ ⌈

• |S| ≥

• γko (T ) = γk (T ) [66] • γko (T ) = n+|L2σ (T )| [67] ⌈ ⌉ • |S| ≥ n(k+2)−3c+1 [43] k+3

1

γko (G) ⌈ ⌉ • δn2+k ≤ γko (G) ≤ ⌈ ⌉ n − δn −k+2 [43,59] 2 • γko (G) ≤ ⌊ ⌋ n(2k+1) [43,59] 2k+2 ⌈ ⌈ ⌉⌉ • µn∗ δn2+k ≤ γko (G) ≤ ⌊ ⌋ γk (G)+n [44,59] 2 ⌈ ⌉ • γko (G) ≥ 2m+kn [43] 3δ +k

)

Kn

P

• a1o (T ) ≤

T

• a2o (G) > • a2o (G) ≤ • a2o (G) ≤

δn +1 o 2 ≤ a1 (G) ≤ γ (G)+n [4] 2 • a1o (G) ≤ 2n 3 [4]

•

G

a2o (G)

a1o (G)

Graph Offensive k-alliance numbers classes

Table 4 Previous results on offensive k-alliance numbers for various graph classes.

K. Ouazine et al. / AKCE International Journal of Graphs and Combinatorics ( 25

• a1o (Pn ) =

Pn

G1× G2

L(G)

• a1o (Cn ) =

Cn

Rδ

a1o (G)

[4]

⌊n⌋

2

2

[4]

⌈n⌉

• a⌋2o (Pn ) = ⌊ n 2 + 1 [4]

• a2o (Cn ) =

a2o (G)

Table 4 (continued) Graph Offensive k-alliance numbers classes

2

⌈n⌉ [4]

[69]

≤ γ1o (Rδ ) ≤ • γ2o (R3 ) ≤

γ2o (G)

• γ1o (G 1 × G 2 ) ≥ 1 o o 2 max{γ (G 1 )γ1 (G 2 ), γ1 (G 1 )γ (G 2 )} [61] • γ1o (B1 × B2 ) ≤ n 12n 2 [61] • γ1o (K r × Pt ) ≥ γ1o (K r )γ1o (Pt ) [61] • γ1o (Cr × Ct ) ≥ γ1o (Cr )γ1o (Ct ) [61] • γ1o (Pr × Ct ) ≥ γ1o (Pr )γ1o (Ct ) [61] • γ1o (K r × K t ) ≥ γ1o (K r )γ1o (K t ) [61]

n 2

•

n(δ+1) 3δ+1

γ1o (G) 4

⌊ 3n ⌋ [60]

• ako (G 1 × G 2 ) ≥ ako1 (G 1 )ako2 (G 2 ) [25]

ako (G)

2

o • γk− ¯ 2 (G 1 × G 2 ) ≥ ∆ n 2 γko (G 1 ) [25]

⌈ ⌉, k=0 [25,68] n3 ⌈ ⌉, k = 1, 2 2 • γko (Pn ) = { n k=0 ⌈ ⌉, [25,68] n3 ⌊ ⌋ + k − 1, k = 1, 2 2 • γko (L(G)) ≥ ⌈ ∑n ⌉ 2 i=1 δi +m(k−2) [43] 3(δ1 +δ2 −2)+k • γko (L(G)) ≥ ⌈ ⌉ n 2(δ−1)+k [43,59] 4 2

• γ o (C ) = { nk n

• γko (Rδ ) = n − α⌊ δ−k ⌋ (Rδ ) [25,68]

γko (G)

26 K. Ouazine et al. / AKCE International Journal of Graphs and Combinatorics ( ) –

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the global offensive k-alliance number γko (L(G)) is investigated. However, for the global offensive 2-alliance number γ2o (G) there is no result in the case of cycle, path and Cartesian product graph classes.

3.2. Study of offensive k-alliance partition numbers Like the partitioning of graphs into defensive k-alliances the partitioning into offensive k-alliances is also studied in the literature. There are two parameters of the partitioning of graphs into offensive k-alliances which are defined go as follows: for any graph G = (V, E), the (global) offensive k-alliance partition number of G, (ψk (G)) ψko (G), k ∈ {2 − δ1 , . . . , δ1 }, is defined to be the maximum number of sets in a partition of V such that each set is (a global offensive) an offensive k-alliance [25]. We say that G is partitionable into (global) offensive k-alliances if go (ψk (G) ≥ 2) ψko (G) ≥ 2. Note that if every vertex of G has even degree and k is odd, or every vertex of G has odd degree and k is even, then every (global) offensive k-alliance in G is an offensive (a global offensive) (k + 1)-alliance go go o and vice versa. Hence, in such a case, ψko (G) = ψk+1 (G) and ψk (G) = ψk+1 (G) [24,25]. In this subsection, we study the mathematical properties of the offensive k-alliance partition numbers. In particular, we exhibit bounds and/or exact values obtained for the (global) offensive k-alliance partition number for some graph classes. 3.2.1. General graphs Sigarreta et al. [24] and Yero [25] studied the partitioning of graphs into (global) offensive k-alliances and obtained several results. Using the relation between the offensive k-alliance number and the offensive k-alliance o partition number (ako (G)ψko (G) ≤ n), they established that lower bounds on ako (G) lead to ⌈ δnupper ⌉ bounds on ψk (G). +k o For example, , they obtained that 2 { from the lower bound given by Fernau et al. in [59], that is ak (G) ≥ ψko (G) ≤

2n ⌋, δn + k 2n ⌋, ⌊ δn + k + 1

⌊

δn + k even, δn + k odd.

. Note that this bound is attained, for instance, for every δ-regular graph, δ ≥ 1,

by taking k = 2 − δ. In such a case, each vertex is an offensive (2 − δ)-alliance and ψko (G) = n as illustrated in [24,25]. Analogously, by using the relation between the global offensive k-alliance number and the global offensive kgo alliance partition number, that is γko (G)ψk (G) ≤ n, Sigarreta et al. [24] and Yero [25] established that lower bounds go on γko (G) lead ⌈ to upper ⌉ bounds on ψk (G). Thus, from the lower bound presented by Bermudo et al. in [68], that is γko (G) ≥ 2m+kn , they obtained that the global offensive k-alliance partition number is bounded upperly by ⌊ 3δ1 +k ⌋ go

ψk (G) ≤

n ⌈

2m+kn 3δ1 +k

⌉

. This bound is attained, for instance, for the circulant graph C R(n, 2) for k = −2, and if

n = 3 j it is also attained for k ∈ {−1, 0} as mentioned in [24,25]. On the other hand, they showed that if a graph G is partitionable into global offensive k-alliances, then ⌊ ⌋ go (i) ψk (G) ≤ 2m−n(k−4) . 2n ⌊ δn −k+4 ⌋ go (ii) ψk (G) ≤ . ⌊ 2 √ ⌋ 4−k+ k 2 +2(δn −k) go (iii) ψk (G) ≤ . 2 ⌊ ⌋ go Sigarreta et al. [24] and Yero [25] also obtained that for any graph G of order n and size m, ψk (G) ≤ 6m+nk , 2m+nk go and they noted that this bound is attained for the circulant graph C R(5n, 2), where ψ−2 (G) = 5. Moreover, they established bounds for the cardinality of sets belonging Thus, that if S belongs to a ⌈ to a partition. ⌉ ⌊ they showed ⌋ 2nδ1 n(2δn −δ1 +k) partition of G into global offensive k-alliances, then δ +2δn +k ≤ |S| ≤ δ +2δn +k . These bounds are reached 1 1 ¯ such for the circulant graph C R(n, 2) which contains a partition into two global offensive 0-alliances S and S, n 2n ¯ that |S| = ⌈ 3 ⌉ and | S| = ⌊ 3 ⌋ [24,25]. Furthermore, they proved that for a graph G with Laplacian spectral radius√µ∗ , if S belongs offensive k-alliances with −δn ≤ k ≤ µ∗ − δn , then ⌈ ⌉ to a partition ⌊ ⌋ √of G into global n 2

−

n 2 (µ∗ −k)−2nm 4µ∗

≤ |S| ≤

n 2

+

n 2 (µ∗ −k)−2nm 4µ∗

. Note that these bounds are attained for the complete graph

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K n with n is even and k = 1. In this case K n is partitioned into two global offensive 1-alliances of cardinality discussed in [24,25].

n 2

as

3.2.2. Cartesian product graphs Let G i = (Vi , E i ) be a graph of order n i , i ∈ {1, 2}. Sigarreta et al. [24] and Yero [25] studied the partitioning of Cartesian product graphs into offensive k-alliances and obtained that if G i = (Vi , E i ) is a graph of minimum ¯ i , i ∈ {1, 2} and Si is an offensive ki -alliance in G i , i ∈ {1, 2}, then, for degree δ¯i and maximum degree ∆ ¯ ¯ k = min{k2 − ∆1 , k1 − ∆2 }, S1 × S2 is an offensive k-alliance in G 1 × G 2 . Thus, they deduced that a partition Πroi (G i ) = {S1(i) , S2(i) , . . . , Sr(i) } of G i into ri offensive ki -alliances, i ∈ {1, 2}, induces a partition of G 1 × G 2 into r1r2 i ¯ 1 , k1 − ∆ ¯ 2 }. This partition is formally illustrated by the following matrix offensive k-alliances, with k = min{k2 − ∆ given in [24,25]: ⎫ ⎧ (1) ⎪ ⎪ S1 × S1(2) · · · S1(1) × Sr(2) ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ S (1) × S (2) · · · S (1) × S (2) ⎪ r2 2 1 2 o . Πr1 r2 (G 1 × G 2 ) = .. .. .. ⎪ ⎪ . ⎪ ⎪ . . ⎪ ⎪ ⎪ ⎭ ⎩ S (1) × S (2) · · · S (1) × S (2) ⎪ r1 r1 r2 1 ¯ i, As a consequence, Sigarreta et al. [24] and Yero [25] established that for any graph G i of maximum degree ∆ o o o ¯ ¯ i ∈ {1, 2}, and for every k ≤ min{k1 − ∆2 , k2 − ∆1 }, ψk (G 1 × G 2 ) ≥ ψk1 (G 1 )ψk2 (G 2 ), and they noted that for the o particular case of the graph C4 × K 4 , ψ−3 (C4 × K 4 ) = 8 = 4 · 2 = ψ0o (C4 )ψ1o (K 4 ). For the global offensive k-alliance partition number, Sigarreta et al. [24] and Yero [25] showed that for G i = go (Vi , E i ) a graph of order n i and Πri (G i ) a partition of G i into ri global offensive ki -alliances, i ∈ {1, 2}, if xi = min S∈Πrgo (G i ) {|S|} and k ≤ min{k1 , k2 }, then i

(i) (ii)

γko (G 1 × G 2 ) ≤ min{n 2 x1 , n 1 x2 }; go go go ψk (G 1 × G 2 ) ≥ max{ψk1 (G 1 ), ψk2 (G 2 )}.

They mentioned that if G i is partitionable into global offensive ki -alliances, for ki ≥ 1 and i ∈ {1, 2}, the bound concerning the global offensive k-alliance partition number is attained for 1 ≤ k ≤ min{k1 , k2 }, where go go go ψk (G 1 × G 2 ) = 2 = max{2, 2} = max{ψk1 (G 1 ), ψk2 (G 2 )}. Moreover, Sigarreta et al. [24] and Yero [25] deduced that if a graph G i of order n i is partitionable into global offensive ki -alliances, i ∈ {1, 2}, then for n1 n2 . Example of equality is obtained in [24,25] for C4 × K 2 , k ≤ min{k1 , k2 }, γko (G 1 × G 2 ) ≤ max{ψ go (G go ),ψ (G )} i.e. γ1o (C4 × K 2 ) =

4.2

go go max{ψ1 (C4 ),ψ1 (K 2 )}

k1

1

k2

2

= 4.

3.2.3. Circulant graphs — C R(n, 2) Let Zn be the additive group of integers modulo n and let M ⊂ Zn such that, i ∈ M if and only if −i ∈ M. A graph G = (V, E) can be constructed as follows, the vertices of V are the elements of Zn and (i, j) is an edge in E if and only if j − i ∈ M. This graph is called a circulant of order n and it is denoted by C R(n, M). The set M is called the set of generators of the circulant graph. With this notation, a cycle graph is C R(n, {−1, 1}) and the complete graph is C R(n, Zn ). Yero [25] in his thesis studied the partitioning of circulant graphs of type C R(n, 2) = C R(n, {−2, −1, 1, 2}) into global offensive k-alliances and obtained some theoretical results. He established two partitions of C R(n, 2) as go follows: if n is even, Π2 (C R(n, 2)) = {{1, 3, 5, . . . , n − 1}, {2, 4, 6, . . . , n}} is a partition of C R(n, 2) into global go offensive 0-alliances, moreover, if n = 4 j, Π4 (C R(n, 2)) = {{1, 5, . . . , n − 3}, {2, 6, . . . , n − 2}, {3, 7, . . . , n − 1}, {4, 8, . . . , n}} is a partition of C R(n, 2) into global offensive (−2)-alliances. Furthermore, Yero [25] considered the circulant graph G = C R(n, 2) and proved that: (i) (ii) (iii) (iv)

Any dominating set in G is a global offensive (−2)-alliance. G is not partitionable into global offensive 3-alliances or global offensive 4-alliances. go go ψ1 (G) = ψ2 (G) = 2 if and only if n = 4 j. go go ψ−1 (G) = ψ ⌋ = 3 if and only if n = 3 j. ⌊ 0 (G) go

(v) ψ−2 (G) = ⌈ nn ⌉ . 5

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go

3.2.4. Relations between ψk (G) and k go Sigarreta et al. [24] and Yero [25] studied relations between ψk (G) and k and they first obtained that, for any graph G without isolated vertices, there exists k ∈ {0, . . . , δn } such that G is partitionable into global offensive k-alliances. As a consequence, they deduced that any graph without isolated vertices is partitionable into global offensive 0-alliances. Furthermore, they showed that if a graph is partitionable into r ≥ 3 global offensive k-alliances, then k ≤ 3 − r . From this latter result, they obtained an interesting consequence which state that if G is partitionable go into global offensive k-alliances for k ≥ 1, then ψk (G) = 2. Moreover, for a graph G without isolated vertices, they deduced that if k ∈ {2 − δ1 , . . . , 0} then the global offensive k-alliance partition number is bounded by go go 2 ≤ ψk (G) ≤ 3 − k. Note that, for the complete graph K n , ψ0 (K n ) = 2 and for the cycle graph C3n , n ≥ 1, go go ψ0 (C3n ) = 3 [24,25]. On the other hand, Yero [25] proved that if G is a graph of order n such that ψk (G) > 2, go o then for every l ∈ {1, . . . , ψk (G) − 2}, there exists a subgraph, G l , of G of order n(G l ) ≤ n − lγk (G) such that go go ψl+k (G l ) + l ≥ ψk (G). 3.2.5. Partition number and chromatic number go Motivated by the lower and upper bounds given for ψk (G) in Section 3.2.4, that is if k ∈ {2 − δ1 , . . . , 0} then go go go 2 ≤ ψk (G) ≤ 3 − k, Sigarreta et al. [24] and Yero [25] studied the limit cases ψ0 (G) = 2 and ψ0 (G) = 3. Essentially, in this study they established relationships that exist between the chromatic number of G, χ(G), and go ψ0 (G). Given a positive integer t, a t-dependent set in G is a set of vertices of G such that no vertex in the set is adjacent to more than t vertices of the set. Thus, a 0-dependent set in G is simply an independent set of vertices in G. Sigarreta et al. [24] and Yero [25] showed that, any set belonging to a partition of a graph into r ≥ 3 global offensive k-alliances is a (−k)-dependent set. Moreover, they noted that if k = 0 in this result, then r = 3 and as a consequence, every set go in a partition into three global offensive 0-alliances is an independent set, and also if ψ0 (G) = 3 then χ(G) ≤ 3. A go go trivial example of graph where ψ0 (G) = 3 and χ (G) = 3 is the cycle graph C3 , and a graph where ψ0 (G) = 3 and χ(G) = 2 is the cycle graph C6 [24,25]. go Sigarreta et al. [24] and Yero [25] also obtained that if G is a non bipartite graph and ψ0 (G) = 3, then χ(G) = 3, go and they mentioned that the complete graph K n with n ≥ 4 is an example of graph where χ(G) > 3 and ψ0 (G) = 2. go Moreover, they deduced that for any graph G without isolated vertices and χ(G) > 3, ψ0 (G) = 2. On the other hand, they have given another sufficient condition for the global offensive 0-alliance partition number to be 2, that go is for any graph G without isolated vertices containing a vertex of odd degree, it is satisfied ψ0 (G) = 2. Sigarreta go et al. [24] and Yero [25] remarked that this latter result is equivalent to saying that if ψ0 (G) = 3, then every vertex in G has even degree. As a consequence of this, for k odd, every partition of G into (global) offensive k-alliances is a go partition of G into (global) offensive (k + 1)-alliances and vice versa. This latter leads to obtain that, if ψ0 (G) = 3 go go o o o o o o and k is odd, then ak (G) = ak+1 (G), γk (G) = γk+1 (G), ψk (G) = ψk+1 (G) and ψk (G) = ψk+1 (G) [24,25]. Now, we summarize the results presented above by giving some bounds obtained for offensive k-alliance partition numbers for general, circulant and Cartesian product graph classes. These results are given in Table 5. Concluding remarks 4. As we can see from Table 5, and comparing with Table 4, we deduce that the offensive k-alliance partition numbers are studied on much less graph classes contrary to the offensive k-alliance numbers. go Furthermore, we note that only the offensive partition numbers with index k namely ψko (G) and ψk (G) that are go investigated. Between these two parameters, the global offensive k-alliance partition number ψk (G) is the most studied one. Moreover, there are only three graph classes which are addressed in this case and the general class is the most studied one.

4. Powerful k-alliances in graphs In this section, we study mathematical properties of powerful k-alliances by giving bounds and/or exact values of several parameters studied for various graph classes. A powerful k-alliance is a set of vertices S ⊂ V of a graph G = (V, E), which is both defensive k-alliance and offensive (k + 2)-alliance. The case k = −1 (resp. k = 0) corresponds to the standard powerful alliances (resp. strong powerful alliances) defined in [1,7]. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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Table 5 Previous results on offensive k-alliance partition numbers for some graph classes. Graph classes

G

Offensive k-alliance partition numbers go

ψko (G)

ψk (G)

• ako (G)ψko (G) ⎧ ≤ n [24,25] ⎨⌊ 2n ⌋, δn + k • ψko (G) ≤ ⎩⌊ 2n ⌋,

• γko (G)ψk (G) ≤ n [24,25] ⌋ ⌊

δn + k + 1

go

δn + keven, δn + kodd.

[24,25]

go

• ψk (G) ≤

n

[24,25] ⌋ go [24,25] • ψk (G) ≤ 2m−n(k−4) 2n ⌊ ⌋ go δn −k+4 • ψk (G) ≤ [24,25] ⌊ 2 √ ⌋ go 4−k+ k 2 +2(δn −k) • ψk (G) ≤ [24,25] 2 ⌊ ⌋ go • ψk (G) ≤ 6m+nk 2m+nk [24,25] ⌈ ⌉ ⌊ ⌋ n(2δn −δ1 +k) 2nδ1 ≤ |S| ≤ δ +2δ [24,25] • δ1 +2δn +k +k ⌈ ⌉ 1 n ⌊ ⌋ √ √ n 2 (µ∗ −k)−2nm n 2 (µ∗ −k)−2nm n n • 2− ≤ |S| ≤ + [24,25] 4µ∗ 2 4µ∗ ⌈

2m+kn 3δ1 +k

⌉

⌊

go

• ψk (G) = 2 [24,25] go • 2 ≤ ψk (G) ≤ 3 − k [24,25] G1 × G2

• ψko (G 1 × G 2 ) ≥ ψko1 (G 1 )ψko2 (G 2 ) [24,25]

G = C R(n, 2)

go

go

go

• ψk (G 1 × G 2 ) ≥ max{ψk1 (G 1 ), ψk2 (G 2 )} [24,25] go

go

• ψ1 (G) = ψ2 (G) = 2 [25] go go • ψ−1 (G) = ψ0 (G) = 3 [25] ⌊ ⌋ go • ψ−2 (G) = ⌈ nn ⌉ [25] 5

Several parameters have been defined and studied in the literature for powerful k-alliances, one can see [6,7,25,40] and others. These parameters are defined as follows: The powerful (−1)-alliance number known as powerful alliance p p number a−1 (G) (resp. powerful 0-alliance number known as strong powerful alliance number a0 (G)) is the minimum cardinality among all (critical) powerful (−1)-alliances (resp. powerful 0-alliances) of G [6,7]. The global powerful p p (−1)-alliance number γ−1 (G) (resp. global powerful 0-alliance number γ0 (G)) is the minimum cardinality among all (critical) global powerful (−1)-alliances (resp. global powerful 0-alliances) of G [6]. The powerful k-alliance number p ak (G) is the minimum cardinality among all (critical) powerful k-alliances of G [43]. The global powerful k-alliance p number γk (G) is the minimum cardinality among all (critical) global powerful k-alliances of G [43]. For any graph G, we have the following relations between different powerful k-alliance numbers: (1) (2) (3) (4) (5) (6)

p

d o a−1 (G) ≥ max{a−1 (G), a−1 (G)} [6]; p p γ−1 (G) ≥ max{γ (G), a−1 (G)} [6]; p o ak (G) ≥ max{akd (G), ak+2 (G)} [43]; p p γk (G) ≥ max{γ (G), ak (G)} [43]; p o γk (G) ≥ max{γkd (G), γk+2 (G)} [43]; p p γk+1 (G) ≥ γk (G) [43].

4.1. Study of powerful k-alliance numbers for various graph classes In this subsection, we exhibit mathematical properties of powerful k-alliances for different graph classes. Essentially, we give bounds or exact values for powerful k-alliance numbers studied for various graph classes. 4.1.1. General graphs In this paragraph, we present theoretical results representing bounds for powerful k-alliance numbers in general graphs. Let G = (V, E) be a general graph of order n and size m. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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For any connected graph G of order n ≥ 2, Brigham et al. [6] studied powerful k-alliances and proposed a sharp upper bound for the powerful (−1)-alliance number. Thus, they used the packing number ρ(G) which is the maximum cardinality of a packing in G (a subset P ⊂ V is called a packing in G if for every vertex v ∈ V , |N [v]∩P| ≤ 1 [6,71]). p They obtained that the powerful (−1)-alliance number is bounded by a−1 (G) ≤ n − ρ(G). Brigham et al. [6] established lower bounds for the global powerful (−1)-alliance number by using the order of G, its maximum degree δ1 , its minimum degree δn and its domination number γ (G). Thus, they showed that for p p n +2 )γ−1 (G) which leads to obtain that γ−1 (G) ≥ δn(δ+δn +1) . Furthermore, any graph G, its order satisfies n ≤ ( δ1δ+δ 1 n +2 ⌊ δn ⌋n +1 p they proved that γ−1 (G) ≥ γ (G) + 2 . Moreover, they obtained a ⌊sharp ⌋ upper bound for the same parameter by p showing that for any graph G with no isolated vertices, γ−1 (G) ≤ n − δ2n . On the other hand, Rodr´ıguez-Vel´azquez and Sigarreta [40] gave tight lower bounds for the global powerful (−1)-alliance number and the global powerful 0-alliance number of the order of a simple graph radius ⌈ √ G, ⌉its size m and its⌈spectral ⌈ by means ⌉ ⌉ λ. Thus, they showed √ ⌈ m+n ⌉ p p p p 1+ 1+8(n+m) 2m+n 2m+n , γ−1 (G) ≥ that γ−1 (G) ≥ 4(λ+1) , γ0 (G) ≥ 2λ+1 and γ (G) ≥ . Note that Rodr´ıguez0 2 4 Vel´azquez and Sigarreta [40] have presented graphs for which these bounds are reached. Fernau et al. [44] studied the powerful k-alliances and ⌈ established lower and upper ⌉ bounds for the global powerful √ ⌋ ⌊ 8m+4n(k+2)+(k+1)2 +k+1 p ≤ γk (G) ≤ n − δn2−k . These k-alliance number. They obtained that for any graph G, 4 bounds are also given by Sigarreta [43] by assuming that k ∈ {1 − δn , . . . , δn − 2} for the upper bound (note that these bounds are reached, for example, for the cycle graph G = C3 for every k ∈ ⌈{−2, −1, ⌉ 0}). Moreover, Brigham and p Dutton [72] obtained a lower bound for the same parameter, that is γk (G) ≥ δ1 +k+1 . By using the spectral radius 2 ⌈ ⌉ p 2m+n(k+2) λ, Sigarreta [43] obtained that for any graph G, γk (G) ≥ 4λ−2k+2 . Yero [25] and Yero and Rodr´ıguez-Vel´azquez [32] studied the mathematical properties of boundary powerful ⌉ ⌊ 2n−δn +k k⌋ ⌈ δn +k+2 ≤ |S| ≤ alliances. They obtained that if S is a boundary powerful k-alliance in a graph G, then 2 ⌉ 2 ⌈ ⌊ ⌋.

Furthermore, they showed that if S is a global boundary powerful k-alliance then 2m+n(k+2) ≤ |S| ≤ 2m+n(k+2) 2δ1 +2 2δn +2 ⌈ ⌉ ⌊ ⌋ n(2δ1 +k+2)−2m n(2δn +k+2)−2m and ≤ |S| ≤ (note that all these bounds are attained, for instance, for the complete 2δn +2 2δ1 +2 graph G = K n for every k ∈ {1 − n, . . . , n − 3}). By using the number of edges of G with one endpoint in S and the other endpoint outside of S, C, they proved that if S is a global boundary powerful k-alliance in G, with k ̸= −1, then |S| = 2(m+n−2C)+nk (see the graph given in Fig. 1(m) of Appendix for illustration, where |S| = 4). 2(k+1)

4.1.2. Tree graphs We present in this part some results concerning powerful k-alliance numbers in trees. Let T = (V, E) be a tree of order n. Brigham et al. [6] initiated the study of powerful k-alliances in graphs and established a sharp upper bound for the powerful (−1)-alliance number in trees. Thus, they deduced from the result given by Meir and Moon in [71] (the p domination number and the packing number of a tree are equal) that for any tree T , a−1 (T ) ≤ n − γ (T ). They also obtained an other ⌋ upper bound for the same parameter, by proving that if T is a tree of order n and T ̸= Pn , ⌊ sharp p . then a−1 (T ) ≤ n+3 2 On the other hand, Rodr´ıguez-Vel´azquez and Sigarreta [45] presented bounds concerning the cardinality of every global powerful (−1)-alliance or 0-alliance in trees. They showed that if S is a global powerful (−1)-alliance ⌈ 3n+8c−2 ⌉ (resp. 0-alliance) in T and the subgraph induced by S has c connected components, then |S| ≥ (resp. 12 ⌈ 2n+4c−1 ⌉ |S| ≥ ). As a consequence, they obtained tight bounds for the global powerful (−1)-alliance number and 5 ⌈ ⌉ ⌈ 2n+7 ⌉ p p the global powerful 0-alliance number, that are γ−1 (T ) ≥ n+2 and γ (T ) ≥ . Rodr´ ı guez-Vel´ a zquez and 0 4 5 Sigarreta [45] have given graphs for which these bounds are attained. Note that these two latter bounds are also given by Sigarreta in his thesis [43]. Sigarreta [43] established a lower bound for the cardinality of every global powerful k-alliance in trees. He showed⌈that if S is⌉ a global powerful k-alliance in T and the subgraph ⟨S⟩ has c connected components, then |S| ≥ n(k+4)+8c−2 (note that this bound is reached, for example, for the graph given in Fig. 1(n) of Appendix, 2(5−k) where |S| = 5). Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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4.1.3. Planar graphs In this paragraph, we put on view bounds obtained for powerful k-alliance numbers in planar graphs. Let P = (V, E) be a planar graph of order n and size m. Rodr´ıguez-Vel´azquez and Sigarreta [45] studied mathematical properties of powerful k-alliances in planar graphs and obtained tight bounds for the global powerful (−1)-alliance number and the global powerful 0-alliance number given as follows: ⌉ ⌈ p . (i) If n > 6, then γ−1 (P) ≥ 2m+n+48 28 ⌈ ⌉ p . (ii) If n > 6 and P is a triangle-free graph, then γ−1 (P) ≥ 2m+n+32 20 ⌈ ⌉ p (iii) If n > 4, then γ0 (P) ≥ m+n+24 . 13 ⌈ ⌉ p (iv) If n > 4 and P is a triangle-free graph, then γ0 (P) ≥ m+n+16 . 9 Furthermore, they showed that if S is a global powerful (−1)-alliance (resp. 0-alliance) in a general graph ⌈ ⌉ ⌈ ⌉ G such that f +16 m+n−4 f +8 the subgraph ⟨S⟩ is planar connected with f faces, then |S| ≥ 2m+n−8 (resp. |S| ≥ ). Moreover, 12 5 Rodr´ıguez-Vel´azquez⌈and Sigarreta in [45] showed that for a global powerful (−1)-alliance (resp. 0-alliance) S in a ⌉ ⌈ m+n+24 ⌉ planar graph, |S| ≥ 2m+n+48 (resp. |S| ≥ ). Enciso and Dutton [33] and Enciso [51] proved that these 28 13 bounds are increased when S is an empire. Thus, they obtained that for a planar graph P with a global powerful ⌈ ⌉ ⌈ ⌉ (−1)-alliance (resp. 0-alliance) S, if S is an empire then |S| ≥ 2m+n+24 (resp. |S| ≥ m+n+12 ). 20 9 Sigarreta in his thesis [43] studied the powerful k-alliances in planar graphs and presented some bounds for the global powerful k-alliance number given as follows: ⌈ ⌉ p (i) If n > 2(2 − k), then γk (P) ≥ 2(m+24)+n(k+2) . 2(13−k) ⌉ ⌈ p . (ii) If n > 2(2 − k) and P is a triangle-free graph, then γk (P) ≥ 2(m+16)+n(k+2) 2(9−k) Moreover, he showed that if S is a global in a general graph G such that the subgraph ⟨S⟩ is planar ⌈ powerful k-alliance ⌉ f +8)+n(k+2) connected with f faces, then |S| ≥ 2(m−4 2(5−k) . Note that Sigarreta [43] have given graphs for which these bounds are attained. Yero and Rodr´ıguez-Vel´azquez [32] studied the boundary powerful k-alliances and proved that if⌊ S is a global ⌈ ⌉ ⌋ f −4 n(k+4)+2 f −4 ≤ |S| ≤ . boundary powerful k-alliance in a planar connected graph with f faces, then n(k+4)+2 2δ1 +2 2δn +2 These bounds are also given by Yero in his thesis [25].

4.1.4. Complete graphs Let K n = (V, E) be a complete graph of order n. We exhibit in this part some exact values obtained for powerful k-alliance numbers in this class of graphs. Brigham et al. [6] studied the powerful k-alliances and they obtained that the powerful (−1)-alliance number and the global powerful (−1)-alliance ⌈ ⌉ number have the same exact value. Thus, they showed that for the complete graph p p K n , a−1 (K n ) = γ−1 (K n ) = n2 . This value is also obtained by Fernau et al. [44] and Sigarreta [43] for the global powerful (−1)-alliance number. Yero in his thesis [25] and Yero and Rodr´ıguez-Vel´azquez [32] studied the boundary powerful k-alliances and ⌉ ⌈ . proved that if S is a boundary powerful k-alliance in a complete graph K n , then |S| = n+k+1 2

4.1.5. Complete bipartite graphs Let K r,s = (X, Y, E) be a complete bipartite graph where r (resp. s) is the cardinality of the set of vertices X (resp. Y ). In this paragraph, we present some results obtained for powerful k-alliance numbers in complete bipartite graphs. Brigham et al. [6] studied the powerful k-alliances in complete bipartite graphs and they obtained a same exact value for the powerful (−1)-alliance number and the global powerful (−1)-alliance ⌊number. ⌋ ⌈ Thus, ⌉ ⌈they ⌉showed that p p s+1 for the complete bipartite graph K r,s , 1 ≤ r ≤ s, a−1 (K r,s ) = γ−1 (K r,s ) = min{r + 2s , r +1 + }. Note that 2 2 Fernau et al. [44] and Sigarreta [43] also established the same value for the global powerful (−1)-alliance number. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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4.1.6. Regular graphs We give in this paragraph some results obtained for powerful k-alliance numbers in regular graphs. We denote by Rδ = (V, E) the δ-regular graph of order n. Brigham et al. [6] studied the powerful k-alliances in regular graphs and they obtained that if S is a powerful (−1)-alliance of a δ-regular graph Rδ , then |∂ S| ≤ |S|. Furthermore, they established that the order of a δ-regular p graph Rδ satisfies n ≤ 2γ−1 (Rδ ), which leads to obtain that the global powerful (−1)-alliance number is bounded by p n γ−1 (Rδ ) ≥ 2 . Yero [25] and Yero and Rodr´ıguez-Vel´azquez [32] studied the boundary powerful k-alliances and showed that if ⌈ ⌉ S is a global boundary powerful k-alliance in a δ-regular graph, then |S| = n(δ+k+2) . They also obtained that if 2(δ+1) f −4 . Rδ is a δ-regular connected planar graph with f faces and S is a global powerful k-alliance, then |S| = n(k+4)+2 2(δ+1) Furthermore, they showed that if S is a global boundary powerful k-alliance in a δ-regular graph Rδ , with k ̸= −1, 2 2 −2k) and C = n(δ +2δ−k , where C the number of edges of Rδ with one endpoint in S and the then |S| = n(δ+k+2)−4C 2k+2 4(δ+1) other endpoint outside of S.

4.1.7. Cycle graphs Let Cn = (V, E) be a cycle graph of order n. In this paragraph, we present some results obtained for powerful k-alliance numbers for the class of cycle graphs. Brigham et al. [6] studied the powerful k-alliances in graphs and they obtained that the powerful (−1)-alliance number and the global powerful ⌈ ⌉ (−1)-alliance number have the same value. Thus, they showed that for any cycle Cn , p p a−1 (Cn ) = γ−1 (Cn ) = 2n . Moreover, Fernau et al. [44] and Sigarreta [43] established the same value written in 3 ⌊ ⌋ p another form for the global powerful (−1)-alliance number, that is γ−1 (Cn ) = n − n3 . 4.1.8. Path graphs Let Pn = (V, E) be a path graph of order n. We exhibit in this part some exact values obtained for powerful k-alliance numbers in path graphs. This class of graphs is studied by Brigham et al. [6] and they obtained that the powerful (−1)-alliance number and the global powerful number are equal in this case. Thus, they showed that for any path Pn , ⌊ (−1)-alliance ⌋ p p a−1 (Pn ) = γ−1 (Pn ) = 2n . Moreover, this exact value, written in another form, is obtained by Fernau et al. [44] and 3 ⌈ ⌉ p Sigarreta [43] for the global powerful (−1)-alliance number, that is γ−1 (Pn ) = n − n3 . 4.1.9. Cartesian product graphs ¯ i , i ∈ {1, 2}. Let G i = (Vi , E i ) be a graph of order n i , minimum degree δ¯i and maximum degree ∆ Yero [25] and Yero and Rodr´ıguez-Vel´azquez [27] studied the powerful k-alliances in Cartesian product graphs and obtained some results for the associated parameters. Thus, they showed that if Si ⊂ Vi is a powerful ki -alliance ¯1 − ∆ ¯ 2 , . . . , min{k1 − in G i , i ∈ {1, 2}, then S1 × S2 is a powerful k-alliance in G 1 × G 2 , for every k ∈ {−∆ ¯ ¯ ∆2 , k2 − ∆1 }}. As a consequence, they obtained that if G i contains powerful ki -alliances, i ∈ {1, 2}, then for every ¯1 −∆ ¯ 2 , . . . , min{k1 − ∆ ¯ 2 , k2 − ∆ ¯ 1 }}, a p (G 1 × G 2 ) ≤ a p (G 1 )a p (G 2 ). Furthermore, they proved that if k ∈ {−∆ k k1 k2 S1 ⊂ V1 is a global powerful k1 -alliance in G 1 , then S1 × V2 is a global powerful k-alliance in G 1 × G 2 , for every ¯ 1 −∆ ¯ 2 , . . . , k1 − ∆ ¯ 2 }. As a consequence, they obtained that if G 1 contains global powerful k1 -alliances, then k ∈ {−∆ ¯ ¯ ¯ 2 }, γ p (G 1 × G 2 ) ≤ γ p (G 1 )n 2 . for every k ∈ {−∆1 − ∆2 , . . . , k1 − ∆ k k1 Remark 5. Let us note that the powerful k-alliances were studied for the class of cubic graphs by Sigarreta in his p thesis [43]. He established some relations between γ (G) and γk (G), k ∈ {−3, −2, −1, 0, 1}, and gave lower bounds p p for γ−1 (G) and γ0 (G). Now, we summarize the results presented above by giving some bounds and exact values obtained for powerful k-alliance numbers for different graph classes. These results are given in Table 6. Concluding remarks 5. As we can see from Table 6, the most studied parameter is the global powerful (−1)-alliance p p number (γ−1 (G)) and the least studied one is the powerful k-alliance number (ak (G)). Furthermore, the general and tree graph classes are the most studied ones and the regular graphs class is the least studied one. Moreover, some p parameters are not studied for all or certain graph classes. For example, the powerful 0-alliance number a0 (G), the Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

[6]

⌊ 2n ⌋

3

δn 2

[6] ⌊ ⌋

≥ γ (G) + [6] ⌊ ⌋ δn ≤ n − 2 [6] ⌈ ⌉ 2m+n ≥ 4(λ+1) [40] ⌈√ ⌉ 2m+n ≥ [40] 2

• •

• •

n 2 [6] ⌈ 2n ⌉ p γ−1 (Cn ) = 3 [6] ⌊ ⌋ p γ−1 (Cn ) = n − n3 [43,44] ⌊ ⌋ p γ−1 (Pn ) = 2n 3 ⌈ [6] ⌉ p γ−1 (Pn ) = n − n3 [43,44]

• γ−1 (Rδ ) ≥

p

• γ−1 (K r,s ) = ⌊ ⌋ ⌈ ⌉ min{r + 2s , r +1 + 2 ⌈ s+1 ⌉ } [6,43,44] 2

p

⌈ ⌉ • |S| ≥ 3n+8c−2 [45] 12 ⌈ ⌉ p • γ−1 (T ) ≥ n+2 [43,45] 4 ⌈ 2m+n+48 ⌉ p • γ−1 (P) ≥ [45] 28 ⌈ ⌉ f +16 [45] • |S| ≥ 2m+n−8 12 ⌈ ⌉ [33,51] • |S| ≥ 2m+n+24 20 ⌈ ⌉ p • γ−1 (K n ) = n2 [6,43,44]

•

•

•

•

p γ−1 (G) p γ−1 (G) p γ−1 (G) p γ−1 (G)

⌉ [40]

⌈ ⌉ • |S| ≥ 2n+4c−1 [45] 5 ⌈ ⌉ p • γ0 (T ) ≥ 2n+7 [43,45] 5 ⌈ m+n+24 ⌉ p • γ0 (P) ≥ 13 ⌉ [45] ⌈ f +8 [45] • |S| ≥ m+n−4 5 ⌈ m+n+12 ⌉ • |S| ≥ [33,51] 9

[40]

m+n 2λ+1

p • γ0 (G) ≥ ⌈ √ ⌉ 1+ 1+8(n+m) 4

• γ0 (G) ≥

⌈

p

• ak (G 1 × G 2 ) ≤ p p ak1 (G 1 )ak2 (G 2 ) [25,27]

p

ak (G)

⌈

n(k+4)+8c−2 2(5−k)

⌉

[43]

p

• γk (G 1 × G 2 ) ≤ p γk1 (G 1 )n 2 [25,27]

⌈ ⌉ p • γk (P) ≥ 2(m+24)+n(k+2) [43] 2(13−k) ⌈ ⌉ 2(m−4 f +8)+n(k+2) • |S| ≥ [43] 2(5−k)

• |S| ≥

p

γk (G) ⌈√ ⌉ 8m+4n(k+2)+(k+1)2 +k+1 • ≤ 4 ⌈ ⌉ p γk (G) ≤ n − δn2−k [43,44] ⌈ ⌉ p • γk (G) ≥ δ1 +k+1 [72] 2 ⌉ ⌈ p • γk (G) ≥ 2m+n(k+2) [43] 4λ−2k+2

)

G1× G2

[6]

3

⌈ 2n ⌉

n(δn +1) δ1 +δn +2

p

• γ−1 (G) ≥

p

p

γ0 (G)

p

γ−1 (G)

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p

• a−1 (Pn ) =

Pn

=

•

p a−1 (Cn )

⌈n⌉

2 [6] p • a−1 (K r,s ) = min{r + ⌊ s ⌋ ⌈ r +1 ⌉ ⌈ s+1 ⌉ + 2 } [6] 2 , 2

p

• a−1 (K n ) =

• a−1 (T ) ≤ n − γ (T ) [6] ⌊ ⌋ p • a−1 (T ) ≤ n+3 [6] 2

p

• a−1 (G) ≤ n − ρ(G) [6]

p

a−1 (G)

p

Powerful k-alliance numbers

Cn

Rδ

K r,s

Kn

P

T

G

Graph classes

Table 6 Previous results on powerful k-alliance numbers for various graph classes.

34 –

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p

upper powerful (−1)-alliance number A−1 (G), the upper powerful 0-alliance number A0 (G) and the upper powerful p k-alliance number Ak (G) are not studied for all graph classes (note that these three latter numbers which are not defined in the literature can be similarly defined as in the cases of defensive and offensive k-alliances). Also, the classes of bipartite graphs and line graphs are not studied for this kind of alliances. In addition, the powerful k-alliance p number ak (G) is studied just in the case of Cartesian product graphs. Besides, in the regular (resp. Cartesian product) p p p graphs class, just the global powerful (−1)-alliance number γ−1 (G) (resp. ak (G) and γk (G)) is (resp. are) studied. p p On the other hand, γ0 (G) and γk (G) are not investigated for several graph classes such as the complete and complete bipartite graphs. 4.2. Study of powerful k-alliance partition numbers Like partitioning of graphs into defensive k-alliances or into offensive k-alliances, the partitioning of graphs into powerful k-alliances is also studied in the literature. There are two parameters of the partitioning of graphs into powerful k-alliances which are defined as follows: For any graph G = (V, E), the (global) powerful k-alliance gp p partition number of G, (ψk (G)) ψk (G), k ∈ {−δ1 , . . . , δ1 − 2}, is defined to be the maximum number of sets in a partition of V such that each set is a (global) powerful k-alliance [25,27]. We say that a graph G is partitionable into gp p (global) powerful k-alliances if (ψk (G) ≥ 2) ψk (G) ≥ 2. In this subsection, we present theoretical results obtained for the powerful k-alliance partition numbers. We give bounds and/or exact values for these parameters in general graphs and Cartesian product graphs, together with some results concerning partitions into boundary powerful k-alliances. 4.2.1. General graphs Yero [25] and Yero and Rodr´ıguez-Vel´azquez [27] studied the partitioning of graphs into global powerful k-alliances and established several results. They showed that, if there are two different sets in Πr (G) (a partition of a graph G into r dominating sets) such that one of them is a defensive k-alliance and the other one is an offensive (k + 2)-alliance, then k ≤ 1 − r . From this result, there are two direct and useful consequences [25,27]: the first one is that for k ≥ 0, no graph is partitionable into global powerful k-alliances; and the second one states that if a graph G is gp partitionable into global powerful k-alliances, then ψk (G) ≤ 1−k. Note that this latter bound is achieved for instance for the complete graph, which can be partitioned into two global powerful (−1)-alliances [25,27]. Furthermore, Yero [25] and Yero and Rodr´ıguez-Vel´azquez [27] proved that for a graph G of order n, ⌊ minimum ⌋ degree δn and gp δ1 +δn +2 maximum degree δ1 , if G is partitionable into global powerful k-alliances then ψk (G) ≤ δn +k+2 . They noted that gp this bound is attained, for instance, for the complete graph G = K n where ψ−1 (G) = 2, or for the circulant graph gp G = C R(3t, 3) for which ψ−4 (G) = 3. Moreover, they obtained an other upper bound in terms only ⌊ √of k and the order ⌋ 8n+(2k−1)2 −2k+1 gp n. Thus, they showed that if G is partitionable into global powerful k-alliances, then ψk (G) ≤ . 4 gp

This bound is attained, for instance, for the circulant graph G = C R(10, 2) for which ψ−4 (G) = 5 as given in [25,27], gp or for the graph given in Fig. 1(o) of Appendix where ψ−1 (G) = 2. 4.2.2. Cartesian product graphs Yero in his thesis [25] and Yero and Rodr´ıguez-Vel´azquez [27] studied the partitioning of Cartesian product ¯ i, graphs into (global) powerful k-alliances. They showed that for a graph G i = (Vi , E i ) of maximum degree ∆ i ∈ {1, 2}, if G i is partitionable into ri powerful ki -alliances, then the graph G 1 × G 2 is partitionable into r = r1r2 ¯1 − ∆ ¯ 2 , . . . , min{k1 − ∆ ¯ 2 , k2 − ∆ ¯ 1 }}. Furthermore, they obtained that powerful k-alliances, for every k ∈ {−∆ p p p ψk (G 1 × G 2 ) ≥ ψk1 (G 1 )ψk2 (G 2 ). Moreover, they established that if G 1 is partitionable into global powerful ¯1 −∆ ¯ 2 , . . . , k1 − ∆ ¯ 2 }, ψ gp (G 1 × G 2 ) ≥ ψ gp (G 1 ), and they remarked that k1 -alliances, then for every k ∈ {−∆ k k1 gp gp if G 1 = C R(3t, 3) and G 2 = K 2 , then ψ−5 (G 1 × G 2 ) = 3 = ψ−4 (G 1 ). 4.2.3. Partitioning a graph into boundary powerful k-alliances Yero [25] studied the partitioning of graphs into boundary powerful k-alliances and he established that every graph can be partitioned into two global boundary powerful (−1)-alliances. Thus he proved that, for a graph G = (V, E): Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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Table 7 Previous results on powerful k-alliance partition numbers for some graph classes. Graph classes

Powerful k-alliance partition numbers p

gp

ψk (G)

ψk (G) gp

• ψk (G) ≤ 1 − k [25,27] ⌊ ⌋ gp n +2 • ψk (G) ≤ δδ1n+δ +k+2 [25,27] ⌊√ ⌋ gp 8n+(2k−1)2 −2k+1 • ψk (G) ≤ [25,27] 4

G

p

p

p

• ψk (G 1 × G 2 ) ≥ ψk1 (G 1 )ψk2 (G 2 ) [25,27]

G1 × G2

gp

gp

• ψk (G 1 × G 2 ) ≥ ψk1 (G 1 ) [25,27]

(i) S ⊂ V is a global boundary powerful (−1)-alliance in G, if and only if, S¯ is a global boundary powerful (−1)-alliance in G. (ii) If G can be partitioned into two global boundary powerful k-alliances, then k = −1. Furthermore, he obtained lower and upper bounds concerning the cardinality of every global boundary powerful (−1)-alliance in terms of the order of the graph G, its minimum degree δ⌈ δ1 .⌋Thus, he n and its⌉maximum degree ⌊ n(δ1 +1) showed that if S is a global boundary powerful (−1)-alliance in G, then δn(δ+δn +1) ≤ |S| ≤ , and he δ1 +δn +2 1 n +2 n noted that if S is a global boundary powerful (−1)-alliance in a δ-regular graph, then |S| = 2 . Moreover, Yero [25] proved that if S ⊂ V is a global boundary powerful (−1)-alliance ⌈ in⌉a graph G =⌊ (V, E) ⌋ and C is a cut set with 2m+n 2m+n one endpoint in S and the other endpoint outside of S, then 2δ ≤ |S| ≤ and |C| = 2m+n . This 2δn +2 4 1 +2 result leads to obtain the previous value of |S| concerning the δ-regular graph. On the other hand, he obtained the result which shows the relationship between the algebraic connectivity of a graph, its Laplacian spectral radius and the respective cardinalities of the two global boundary powerful (−1)-alliances S and S¯ which form a partition of the graph. is a global boundary powerful ⌈√Thus, he proved ⌉ that if S ⊂ V⌊√ ⌋ ⌊√ (−1)-alliance ⌋ in G, then without ⌈√ loss of generality, ⌉ n 2 (µ−1)−2nm n 2 (µ∗ −1)−2nm n 2 (µ∗ −1)−2nm n 2 (µ−1)−2nm n n n n ¯ + ≤ |S| ≤ 2 + and 2 − ≤ | S| ≤ 2 − , where 2 4µ 4µ∗ 4µ∗ 4µ µ (resp. µ∗ ) is the algebraic connectivity (resp. the Laplacian spectral radius) of the graph G. Recently, Slimani and Kheddouci [26] have introduced a new concept of saturated vertices and studied the saturated boundary k-alliances in graphs. They have proved that S ⊂ V is a minimal global boundary powerful (−1)-alliance in G, if and only if, S¯ is a minimal global boundary powerful (−1)-alliance in G. Furthermore, as a main result, they have obtained tight bounds for the cardinality of every minimal global boundary powerful (−1)-alliance in terms only of the order and the size of graph by taking the two cases where ⟨S⟩ is connected or not. Hence, they showed that for a graph G = (V, E) with |V | = n and |E| = m, if S ⊂ V is a minimal global boundary powerful (−1)-alliance, then: (i) If S is connected, one has: { max

−1 +

} { } √ √ √ 1 + 4n 1 + 1 + 8m 2n + 1 − 4n + 1 m + 3 , ≤ |S| ≤ min , . 2 4 2 4

(1)

(ii) If S is not connected, the relation becomes: { max

1+

} { } √ √ √ 4n − 7 5 + 8m − 7 2n + 1 − 4n + 1 , ≤ |S| ≤ min ,m . 2 4 2

(2)

Note that several examples have been presented in [26] for which these bounds are reached. For instance, all the bounds given in (1) are attained at the same time for the complete graph K 2 , and the upper bound m given in (2) is ¯ reached when the graph G is constituted of not adjacent edges and every edge links a vertex of S with a vertex of S. Now, we summarize the results presented above by giving some bounds obtained for powerful k-alliance partition numbers in general and Cartesian product graphs. These results are given in Table 7. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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Concluding remarks 6. As we can see from Table 7, and comparing with Table 6, we deduce that the powerful k-alliance partition numbers are studied on much less graph classes contrary to the powerful k-alliance numbers. gp p Furthermore, we note that only the powerful partition numbers with index k namely ψk (G) and ψk (G) that are gp investigated. Between these two parameters, the global powerful k-alliance partition number ψk (G) is the most studied one. Moreover, there are only two graph classes which are addressed in this case, for the general graphs only gp p gp ψk (G) is studied and for the Cartesian product graphs both ψk (G) and ψk (G) are studied.

5. Conclusion and discussion Since the beginning of the last decade, when alliances in graphs were first introduced, much research has been focused on studying mathematical properties of various parameters of different types of k-alliances in graphs. In this paper, we have surveyed and discussed the principal known results obtained on defensive, offensive and powerful k-alliances by classifying them according to the different graph classes where the parameters are investigated. From this survey, we draw the following conclusions: • By considering the classification criterion “graph class” in the study of the three kinds of k-alliances, we deduce that: the most studied graph classes on which there are more results are general, tree, planar and cartesian product graphs, and the least studied graph classes on which there are fewer results are cycle, path and line graphs. • Several k-alliance numbers are defined in the literature. Some of them are received more attention and have p d been studied for various graph classes, such as γ−1 (G), γko (G) and γ−1 (G). However, there are some parameters p o o which are not studied for all graph classes, such as A1 (G), A2 (G) and a0 (G), and other ones are not studied for p d o certain graph classes, such as Ak (G), ak (G) and ak (G). • The k-alliance partition numbers have been studied on much less graph classes contrary to the k-alliance numbers. Moreover, only the partition numbers with index k are investigated in the case of partitioning of graphs into offensive (powerful) k-alliances. • There are more studies and hence more results obtained on defensive k-alliances than on offensive (powerful) k-alliances. • Some relationships are established between the global offensive k-alliance partition number and a coloration parameter namely the chromatic number. In this sense an extensive study which includes other parameters can be interesting. • There are many investigations in the sense of theoretical aspects of k-alliances, but there are several prospects and progress to carry out in the algorithmic and computational side. • The alliances with their important properties are used in interesting applications in several areas. As prospects, in practice there are many problems which have specific structures where the mathematical properties of the alliances can be involved and contribute to solve these problems. • The definition of the defensive (−1)-alliance which takes into consideration the defense of a single vertex is generalized by Brigham et al. [53] to the concept of secure sets in order to forestall any attack on the entire alliance or any subset of the alliance. In this sense, it would be interesting to consider k-secure sets as extensions of defensive k-alliances and also to study the partitioning of graphs into k-secure sets. In this setting, motivations with practical examples would be needed. Acknowledgments The authors are grateful to all the anonymous referees of the previous versions of the paper for their valuable suggestions and comments which have served to improve its quality. Appendix Note that the graphs given in Fig. 1 are denoted G 1 , G 2 , . . . , G 15 respectively. Please cite this article in press as: K. Ouazine, et al., Alliances in graphs: Parameters, properties and applications—A survey, AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.05.002.

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(a) S1 = {1, 2, 3, 4} is a (strong) global defensive alliance in (G 1 −{e}) G1.

(d) S4 = {1, 2, 3, 4} is a global boundary defensive (resp. offensive) 0-alliance (resp. (−1)alliance) in G 4 .

(g) S7 = {3} is a global defensive (−4)alliance in G 7 .

(j) S10 = {2, 3} is a global offensive 1-alliance in G 10 .

(m) S13 = {2, 5, 7, 8} is a global boundary powerful (−2)-alliance in G 13 .

(b) S2 = {1, 3, 5, 8} is a (global) offensive alliance in (G 2 − {10}) G2.

(e) S5 = {1, 2, 5} is a boundary offensive 1-alliance in G5.

(h) S8 = {1, 2, 6} is a defensive 0alliance in G 8 = L(K 4 ).

(k) S11 = {3, 4, 5} is a global offensive 1-alliance in G 11 .

(n) S14 = {1, 4, 5, 6, 10} is a global powerful (−1)-alliance in G 14 .

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(c) S3 = {1, 3, 6} is a global powerful alliance in G 3 .

(f) S6 = {2, 3} is a global boundary powerful (−2)-alliance in G 6 .

(i) {S9 = {1, 3}, S9′ = {2, 4}} is a partition of G 9 into two defensive (−1)-alliances.

(l) S12 = {2, 4, 6} is a global offensive k-alliance in G 12 = L(C6 ) with k ∈ {1, 2}.

′ = (o) {S15 = {1, 2, 3}, S15 {4, 5, 6}} is a partition of G 15 into two global boundary powerful (−1)-alliances.

Fig. 1. Examples of k-alliances in graphs.

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